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AS Level Mathematics 9709 | Circular Measures 4 | Mensuration & Radians | Raja Shafqat — Transcript

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0:08Assalamu Alaikum, beta. In the last
0:11class, we studied angle properties and
0:13properties of triangles. Now, for
0:15circular measurement, the fifth point
0:18required is O-Level Maths carry forward
0:20, and that point is named mensuration.
0:22Is that clear, beta? So, finally, we
0:24are just doing the last two points of
0:26O-Level Maths. Then, InshaAllah, we
0:28will finally start our chapter. So
0:30today, the mensuration we require is
0:32the mensuration of circles. That is,
0:35the mensuration of the circle family is
0:37required. Number one, in O-Level Maths,
0:39we must have studied what we call this
0:41shape? A circle. Okay? This is
0:43obviously the center of the circle. The
0:46gap from the center to the
0:47circumference is what we call the
0:49radius in maths. And what do we call
0:51the double of the radius? The double of
0:53the radius is what we call the diameter
0:55. So obviously, this means this is also
0:57a radius and this is also a radius. Is
0:59that clear, beta? And what does the
1:01word diameter mean? Diameter is twice
1:03the radius. Meaning the line that
1:06passes through the center, so that
1:08means it is equal to what? Two times
1:10the radius. So that means what is the
1:12formula for diameter? What will the
1:14diameter be? 2r, two times the radius.
1:17What word do we use for the boundary of
1:19a circle? For the boundary of a circle,
1:21we must have heard the word in O-Level
1:23Maths. That word is circumference. The
1:26word we use for the boundary of a
1:28circle is circumference. So we learned
1:30in O-Level Maths what the area of any
1:32circle is? It's always π*r², π*
1:35radius squared. And the circumference
1:38of a circle, what is the formula for
1:40circumference? 2π*r, 2*π*r, this is
1:42the basic formula for circles we study
1:45for the first time in class seven or
1:48eight. So this is the first member of
1:50the circle family. We call the second
1:52member of the circle family a
1:54semi-circle. A semi-circle. Well, what
1:58does the word semi-circle mean? The
2:00word semi-circle means half circle.
2:03Okay? So, we are now talking about a
2:05half circle. So what does the word
2:06semi-circle mean? Half of a circle, or
2:08you could say half circle. In O-Level
2:11Maths, we used to call a half circle. A
2:13semi-circle. A semi-circle. Look, we
2:16studied in O-Level Maths. What do we
2:18call this complete shape? We say this
2:21complete shape is our circle. Is that
2:23clear, beta? This is our circle, and
2:25what do we call the line through the
2:26center? Diameter. And obviously, what
2:29is this? The center of the circle. So,
2:31in the language of math, what do we
2:33call this shape? This shape we have is
2:36a circle. Clear? Now, if I cut this
2:39circle in half. Right? If I cut this
2:41circle in half, what word do we use in
2:43math for that? Semicircle. Look, this
2:46also means that this is your semicircle
2:49. The upper half, and this is also your
2:51semicircle. The lower half. So what are
2:53these two? These both are semicircles.
2:55So what does the word semicircle mean?
2:58The word semicircle means half of a
3:00circle. Half of a circle. So if we are
3:03asked, what is the area of a semicircle
3:05? What would the area of a semicircle
3:07be? So we learned in O-Level math, what
3:09is the area of a semicircle? Take the
3:11complete area of the circle and divide
3:13it by two. Meaning, when we divide any
3:16circle into two equal parts, the half
3:17portion is what we call a semicircle.
3:20So we divided the area into two equal
3:22parts. And if we are asked, what is the
3:25circumference of a semicircle? Or
3:27whether it's this semicircle or that
3:30one, obviously both are semicircles.
3:32They are half circles. So if we are
3:34asked, what is its circumference? The
3:37formula for circumference is 2πr/2. So
3:43when the two and two cancel out, what
3:45answer for circumference is derived? π
3:48*r. So what is the circumference of a
3:50semicircle? π*r. Similarly, we used to
3:53hear another word in O-Level math that
3:55this is a radius and this is a radius,
3:57and this boundary is what we call arc
3:58length or circumference. So if I am
4:01asked, what would be the perimeter of
4:03this entire semicircle? So what does
4:05the word perimeter mean? The word
4:07perimeter means the sum of all sides.
4:10So what will the perimeter be? The
4:11perimeter will be r + r + circumference
4:14. So, it's r + r + πr or if I make it
4:18easier, I can also say diameter plus
4:21circumference. Diameter plus
4:24circumference, meaning this r and this
4:26r, so what will this entire length be?
4:28It becomes the diameter, and this
4:30boundary, what do we call this boundary
4:31? We call this boundary the
4:33circumference. So, if we add both these
4:36sides, both lengths, meaning if we add
4:38the circumference as well as the
4:39diameter, then what will we find? The
4:42perimeter of a semi-circle. We studied
4:45this second shape in O-Level Math. In
4:48O-Level Math, we used to study another
4:50shape which we called a quadrant.
4:52Quadrant, and what word did we use for
4:54a quadrant in O-Level Math? By the way,
4:57what exactly is a quadrant? In the
4:58language of math, we define a quadrant
5:01as 1/4 of a circle. We refer to 1/4 of
5:04a circle as a quadrant in math. That is
5:07, if any circle is divided into four
5:09equal parts, what will each of its
5:11parts be called? Each part is a
5:13quadrant. So, what did we call 1/4 of a
5:16circle in O-Level Math? We used to call
5:19it a quadrant in O-Level Math. That is,
5:23if you have a circle and I divide this
5:26circle into four equal parts, then what
5:29will we call these four parts? We will
5:33call these four parts quadrants. Or
5:35look, I will show you by breaking it
5:38down. I have copied the same circle.
5:40Right? Right? I will show you again and
5:42again by breaking down that same circle
5:43. So, what is a quadrant? Obviously,
5:451/4 of a circle is a quadrant. 1/4 of a
5:48circle is a quadrant. Now, if I show
5:50all these circles like this. Look, if I
5:53show you only this portion of the
5:54circle, then this portion is obviously
5:56what? This is also a quadrant. Or if I
5:58show you this portion of the circle,
6:00then what will this portion be?
6:02Obviously, it will be a quadrant. This
6:04is also a quadrant. Or if I show the
6:06circle like this, then whatever 1/4
6:08part of the circle there is, what did
6:10we call that 1/4 part in O-Level Math?
6:12A quadrant. This is also a quadrant.
6:14And similarly, if I show this
6:16particular portion of the circle, then
6:18what will this part of the circle be?
6:20This part will also be a quadrant. So,
6:22if I combine these four pieces, what
6:25shape will be formed again? You will
6:28have the circle plotted again. Is that
6:31clear, son? You will have the circle
6:33part again. So, what did I do? I
6:35divided the circle into four equal
6:37parts. I divided the circle into four
6:41equal parts. So, what will we call
6:44these four parts? Quadrants. All these
6:47four parts are quadrants of this circle
6:49. Clear, kid? All these parts are its
6:52quadrants. So when we talk about any
6:54one quadrant, like let's suppose this
6:56is one quadrant. Then it means that the
6:58angle of the quadrant will always be
6:59what? It will be 90. Because this
7:02complete angle of 360 was divided into
7:05four equal parts. It was divided into
7:08four equal parts. So that means this
7:10angle is also 90. This angle is also 90
7:11. This one is 90 too, and this one is
7:13also 90. So it means the angle inside
7:15the quadrants is always 90 degrees. So
7:17if we are asked, what will be the area
7:18of the quadrants? Then obviously, what
7:20will be the area of the quadrant? The
7:22area of the quadrant will be the area
7:23of the whole circle divided by 4.
7:25Because a quadrant means you have
7:27divided a circle into four equal parts.
7:30And similarly, if I ask you what this
7:31boundary is, what is this boundary?
7:34Then for its boundary, the term arc
7:36length is also used in math. Arc length
7:39, or you could also say the
7:41circumference of the quadrant.
7:43Circumference of the quadrant. So how
7:45will this be found? We divided the
7:47complete circumference by four. Because
7:49the entire circumference of the circle
7:51is divided into four equal parts. So it
7:54means now if I talk to you about what
7:56its circumference is? Then you would
7:59say πr/2, meaning it will be half of
8:01the semi-circle. So it will be half of
8:04the semi-circle. If we require the
8:06perimeter, what does the word perimeter
8:08mean? r + r + arc length. r + r + arc
8:12length. So if we are asked what the
8:15perimeter of this shape will be? Then
8:17what should the perimeter of this shape
8:19be? Perimeter is r + r + πr/2, clear?
8:24Add these three sides together. Then
8:26what will you find? This is the radius.
8:29And this boundary, this small part, is
8:31also called arc length in math. And arc
8:34length is also denoted by 's' in math.
8:37Clear, kid? Arc length is also denoted
8:39by 's' in math. So what does a quadrant
8:42mean? When we divide a circle into four
8:45equal parts. Then every part of the
8:47circle, or every piece, what is it read
8:49as individually? What do we read it as
8:51individually? Sector, sorry, we will
8:53study sectors later. What is a sector?
8:55We read it as a quadrant. The quadrant
8:58is 1/4, 1/4, 1/4, 1/4; so what is a
9:00quadrant? Basically, a quadrant is 1/4
9:03of a circle. Then we used to hear
9:05another word related to this topic in O
9:08Level Maths. That word was segment.
9:12That word was segment. We also studied
9:14what a segment is in circle properties.
9:17Right? So let me explain it in a bit
9:18more detail. We also studied it in O
9:20Level Maths. We also studied in O Level
9:23Maths what a chord is. Any line that
9:26internally cuts the circle at two
9:28points. In mathematical language, we
9:31call that line a chord. Obviously, a
9:33chord does not pass through the center.
9:35The center of the circle must be
9:37somewhere around here. This is the
9:38center of the circle. The line through
9:40the center, we call that a diameter.
9:42Otherwise, the line that internally
9:44cuts the circle at two points, we call
9:46that a chord of a circle in math terms.
9:49So what does a chord do? Your chord
9:51divides the circle into two unequal
9:53parts. This and this are both unequal
9:55parts. What does a diameter do? The
9:57diameter divides it into two equal
9:58parts. Meaning a semicircle and a
10:00semicircle. So a chord always divides
10:03the circle into unequal parts. So if we
10:06are asked what the definition of a
10:07segment is? In mathematical language, a
10:11segment is the area between a chord and
10:14the circumference. The area between the
10:18chord and the circumference. This is
10:20what we call it in mathematical
10:21language. We call this a segment in
10:23mathematical language. So the area
10:25between the chord and the circumference
10:27is a segment. So we studied in O Level
10:29Maths that this major portion, meaning
10:31the chord divided the circle into
10:33unequal parts. So this upper portion,
10:36we kept calling it a major segment in O
10:38Level Maths. We kept calling it a major
10:41segment in O Level Maths. And this
10:44lower portion, this small piece, the
10:47lower part, we kept calling it a minor
10:49segment in O Level Maths. We called it
10:52minor because it is a small portion,
10:54and major because the upper portion is
10:56a larger portion. So, minor means small
10:59and major means large. So that means
11:01what does a segment do? What is a
11:03segment? The definition of a segment is
11:05the area bounded between a chord and
11:07the circumference. Meaning the area
11:08above the chord and the area below the
11:10chord. The area between the chord and
11:13the circumference is what we read as a
11:15segment. So this is a major segment and
11:17we used to read this in O Level Maths
11:19as a minor sector. A chord always
11:22divides a circle into unequal parts. A
11:26chord always divides a circle into
11:28unequal parts. And a diameter always
11:31divides a circle into equal parts. This
11:33is a very basic thing, by the way.
11:35Clear, kid? It is a very simple matter.
11:37So, so far we have looked at a circle.
11:40Then we looked at a semicircle and then
11:41we looked at a quadrant. What is a
11:43quadrant? A quadrant is 1/4 of a circle
11:46. Meaning, if a circle is divided into
11:49four parts, what do we call each part?
11:51Each part? We call each part a quadrant
11:54. A quadrant is 1/4 of a circle. 1/4 of
11:58a circle. Okay? Now let's continue with
12:00this. Up until here, we studied
12:02segments. Now, one more small point. In
12:05O-Level Math, we used to call this
12:06point a sector. Now look, if I were to
12:09define what a sector is? Look, we
12:12learned in O-Level Math what a sector
12:15is. A sector is the area between two
12:20radii or two radii and an arc length.
12:25The area between two radii and an arc
12:27length. We call this a sector in the
12:30language of math. The area between two
12:32radii or two radii and an arc length.
12:35In math, we call this a sector. Look,
12:37for example, suppose this is a diagram.
12:40Okay? Suppose you have this circle.
12:43Suppose you have this circle. And this
12:45is the center of the circle. The center
12:47of the circle. So when we generate
12:50these two lines from the center of the
12:53circle to the circumference. Then we
12:56used to call this a radius. So what is
12:57this length? It is your radius. This
12:59length is your radius. So, we learned
13:01in O-Level Math that this little angle,
13:04this upper angle—if I label this
13:06upper angle as theta. Then we used to
13:09say in O-Level Math, even in pie charts
13:11, that the lower angle would be what?
13:13360-theta. 360-theta. So in O-Level
13:19Math, we used to define it like this.
13:22Watch closely, kid. I will copy this
13:25same sector again. Okay? Okay? I will
13:28copy this sector again and then explain
13:29to you the minor and major sectors,
13:31what a minor sector is and what a major
13:33sector is. Okay? I have copied this
13:35sector, this circle. Now I have copied
13:38this circle twice. Okay, kid? Wait for
13:421 minute. Okay? I will show you by
13:50copying this sector twice. So when I
13:52copied it twice. Then the lower portion
13:58and the upper portion. Your circle, I
14:05don't know what the issue is. Wait,
14:07look here, look at the same diagram
14:10again. So, this small upper portion,
14:13this upper portion has a smaller angle.
14:16Its angle is theta. This angle here is
14:19theta. And these are your radius and
14:21radius. So, this upper portion, if I
14:25erase this, we used to call this 'minor
14:27sector' in O Level Maths. What is this?
14:31It is the minor portion of the circle.
14:35What did we call this in O Level Maths?
14:38It is a minor sector, meaning the
14:42smaller part of the circle. Similarly,
14:47if I show you the sector again, I
14:49explained above that the angle at a
14:51point is always 360. Okay? So, if I
14:55erase this smaller portion. Look, I
14:57have erased the smaller sector. So,
15:00when I erased the smaller sector, this
15:03lower portion, this portion, we used to
15:05call it the major sector in O Level
15:07Maths. Major sector, this is the radius
15:11. This is the radius. And its angle was
15:14what we called 360-theta. 360-theta. So
15:18, what word did we use for this sector
15:21in O Level Maths? We used the O Level
15:24Maths term. Major sector, major and
15:27minor. It is minor because the minor's
15:30angle is smaller. And it is major
15:32because the major's angle is larger. So
15:34, these were both sectors. In O Level
15:36Maths, we learned what we call both of
15:38these? We call them sectors. Minor and
15:40major, smaller and larger. Also, look
15:43at another interesting thing we studied
15:46in O Level Maths about the connection
15:48between a sector and a circle. What is
15:52the connection of a circle with a
15:54sector? We know that whether it is
15:56minor or major. You must have studied
15:59in the pie chart topic that if the
16:00sector's angle is theta, what is the
16:02angle of a complete circle? 360. So,
16:05basically what is a sector? Whether it
16:07is minor or major. A sector is a part
16:10of a circle. Now look, this entire
16:12portion is also a major sector. It is
16:14also a part of a circle. So, basically,
16:16what is the relation between a sector
16:18and a circle? Angle divided by 360.
16:21Angle divided by 360. If the sector is
16:24minor, then it will be the minor angle.
16:27And if you want to see the relation for
16:28the major sector, you will use the
16:30major sector's angle here. Clear, son?
16:32So, if I summarize this further, I can
16:35also show it like this: what is the
16:36sector equal to? The sector is theta/
16:40360 of the whole circle, of the whole
16:45circle. We learned in O-Level Math what
16:49a sector is. Theta/360 of the whole
16:51circle. So, what is a sector basically?
16:54It is a fraction. A sector is basically
16:57a fraction of a circle. A fraction of a
17:00circle. Clear, kid? A fraction of a
17:03circle. So, this sector, what is every
17:05sector always? If I define a sector, I
17:08can say it is the angle of a sector
17:10divided by the total angle multiplied
17:13by the whole circle. This is the
17:15relationship between a sector and a
17:17circle. Between a sector and a circle.
17:20Okay, kid? Between a sector and a
17:22circle. So, if I talk to you about a
17:24minor sector. Okay? Let's talk about
17:27the minor sector for a minute. Minor,
17:30which portion is the minor one? Minor
17:32is this sector of yours. This, this is
17:35your minor sector. Minor means the
17:37smaller portion. So, if I ask you what
17:39the area of the minor sector would be?
17:41What did we learn in O-Level Math? The
17:44area of the minor sector will be theta/
17:47360 of the whole area. Meaning, of the
17:51circle, right? So, what is the area of
17:52a circle? πr² and if I ask you what
17:55its arc length is? We learned in
17:57O-Level Math that the word
17:59circumference is not used for a sector.
18:02This boundary of the sector or this
18:04boundary of the sector is called arc
18:06length in mathematical language. We
18:09used to call it arc length in O-Level
18:11Math. Arc length. So, what formula did
18:13we learn for arc length? Theta/360 of
18:17the whole circumference. Of the whole
18:20circumference. So, what is the area?
18:22Theta/360 of the whole circle. And what
18:25is the arc length? Theta/360 of the
18:27whole circumference. Meaning, area is
18:30dependent on area. And arc length is
18:33dependent on circumference. Arc length
18:36is dependent on circumference. So, we
18:38used to represent arc length with 's'
18:41in O-Level Math. And obviously, we used
18:43to represent area with 'A'. So, this
18:46sector you have, the minor sector, what
18:49is its area? Theta/360 of the whole
18:52circle. And what is its arc length?
18:55Theta/360 of the whole circumference.
18:58If I ask you what the perimeter of the
19:00minor sector would be? Perimeter. So,
19:03the word perimeter means the sum of all
19:05sides. This side is R, this side is R.
19:07And this boundary, this boundary we
19:09call the arc length. So, I can express
19:12the perimeter like this. What is the
19:13perimeter? r + r + S. So, I can show it
19:17like this. I already know the formula
19:20for 2R and arc length. This is the arc
19:23length formula. The arc length formula
19:27was theta/360*2 π r, so we discussed
19:31the minor sector. Minor, meaning the
19:35smaller sector. Similarly, if I talk to
19:38you about the major sector. Similarly,
19:40if I talk to you about the major sector
19:42. What did we read this sector as? We
19:44read this sector as our major sector.
19:47This sector is our major sector. There
19:50it is, this major sector. We talked
19:52about the minor sector 2 minutes ago.
19:53Now, if we talk about the major one,
19:56obviously, whether it's minor or major,
19:58both are sectors. So when we talk about
20:01the major sector, the major sector. So
20:04look, what is the angle of the major
20:06sector? The angle of the major sector
20:09is 360-theta. So if I ask you what the
20:11area of the major sector will be? Then
20:13you will say the angle of the major
20:16sector. The angle of the major sector.
20:18What is the angle of the major sector?
20:21360-theta over 360 of the whole circle.
20:25Clear, beta? Basically, when it's minor
20:27, the minor angle will be used. When
20:29it's major, the major angle will be
20:31used. Clear, beta? And similarly, if we
20:33talk about arc length or major arc
20:35length. What word did we use for its
20:38arc length in O-level math? We used
20:40this word for it. Major arc length. We
20:42used to read the smaller one's arc
20:44length. Minor arc length, minor arc
20:47length. So what is the relationship of
20:50arc length? The arc length relationship
20:52is with the circumference. So we will
20:54say what is the theta angle? Theta 360-
20:57theta over the total angle of the whole
21:01circumference. Before this, when we
21:05talked about the minor, it was the
21:06minor's angle. Minor angle over 360 of
21:09the whole circumference, major angle
21:11over 360 of the whole circumference. So
21:14what did we read this as? We used to
21:16read this as major arc length. This is
21:18all O-level math. So if I ask you, what
21:20will be the perimeter of this major
21:22sector? Perimeter meant the sum of all
21:25sides. So we will say, what does the
21:27word perimeter mean? The word perimeter
21:29means the sum of all sides. So what
21:31will the perimeter be? r + r + major
21:34arc length. Look, r, r, and this arc
21:37length, what will we read this arc
21:39length as? We used to read this arc
21:41length as major arc length. So I can
21:43summarize it and show it like this. r +
21:45r, and what was the formula for major
21:48arc length? θ/360 of the whole
21:52circumference. So, this was the sector
21:57that students study in ninth-grade
21:59mensuration. Is that clear, son? Now,
22:02whether it is a minor sector or a major
22:03sector, both are sectors. So, what is a
22:05sector? Basically, the area between two
22:08radii and an arc length. That is, the
22:11area bounded between two radii and an
22:13arc length is called a sector in math.
22:17So, this is basically a sector, and
22:19this is also basically a sector. This
22:21was the minor, and the bottom portion
22:23was the major sector. Minor and major.
22:26Clear, son? We also include this in
22:28O-Level math under mensuration of
22:31circles. So, this is also part of the
22:33circle family, which we call a segment.
22:35I mean, we studied the circle, we call
22:37it a semi-sector. We studied the circle
22:40, semi-circle, quadrant, segment, and
22:43sector. Clear, son? Now, finally, look
22:47at one most important thing. Look at
22:49one most important thing. Now, the last
22:51point of this family. Now we are
22:53talking about the segment. Clear, son?
22:55Now we will talk about the segment.
22:57Okay? Because the segment is very
22:59important and is used very frequently
23:02in A-Levels. Now let us look at the
23:04segment in detail. We previously
23:06studied what a segment is. Area bounded
23:09between a chord and the circumference.
23:11Meaning, a chord always divides a
23:12circle into unequal parts. So, the area
23:14between the chord and the circumference
23:16is what we call a segment. Look here
23:19and check it once again, son. We
23:21studied previously today that, suppose
23:23you have a circle, and this is a chord.
23:26We studied today that what do we call
23:28this portion? In the language of math,
23:31we read this as a major segment. A
23:33major segment, and the lower portion,
23:35this one, what do we read this as? In
23:39the language of math, we read this as a
23:42minor segment. Major segment and minor.
23:46Major and minor. Okay, now we are
23:49interested in the segment. Okay? Now,
23:51please look at this very carefully. I
23:53am telling you something new. Now we
23:54are interested in the segment. Suppose
23:57you have this circle. Suppose, and you
24:00have this chord. This is your chord.
24:03Clear? Now, suppose you are interested
24:06in this portion which I am highlighting
24:08. Minor segment. You need to find the
24:11area of the minor segment. The area of
24:13the minor segment. So, what did we
24:15study in O-Level math? That whenever
24:17you require the area of the minor
24:19segment, what did we do? What do we do
24:21in O-Level math? We join both points of
24:25the chord to the center. We join both
24:28points of the chord to the center.
24:30Let's do one thing. Let's label this O,
24:32this A, and this B. So OA is the radius
24:35and OB is also the radius. And what is
24:37AB basically? I will call AB a chord.
24:40AB is a chord. Right? What is AB? AB is
24:43a chord, and let's call this angle
24:45theta. Like we used to call it theta in
24:48O-Level Math. So basically, if you
24:50notice, this whole shape becomes a
24:52sector, and what does this portion
24:55become? An isosceles triangle. An
24:57isosceles triangle. Clear, kid? We will
25:00call this an isosceles triangle. Clear?
25:03Okay. Now, what did we study in O-Level
25:05Math? Right? What did we study in
25:07O-Level Math? That whenever we talk
25:10about the area of a segment. Right? I
25:13will show you again. Whenever we talk
25:15about the area of a segment, how is the
25:17segment's area found? I have copied
25:21this portion here. Copied this portion
25:24here. Right? So what is this? It's your
25:27R. What was R? R is your radius. R for
25:30radius. This is your radius. Suppose
25:34this was your center of the circle, O.
25:35What was this? Center of the circle, O.
25:38And this was your chord. The name of
25:40the chord was AB. And your interest is
25:42in this portion, which I am
25:43intentionally highlighting. Meaning the
25:46minor segment; your interest is in this
25:49minor segment. So what did we study in
25:53O-Level Math, that if you want to find
25:56the area of the minor segment. If you
25:59want to find the area of the minor
26:01segment. Then what did we used to do in
26:03O-Level Math? What did we used to do in
26:06O-Level Math, that this whole shape is
26:07a sector. This entire shape is a sector
26:10. So what did we use to do? We used to
26:13subtract the isosceles triangle from
26:16the sector. Meaning, what we did was
26:19remove this triangle from this entire
26:21shape. Now, if we remember, what was
26:24the area of a sector? We studied in
26:26O-Level Math that the area of a sector
26:30was: theta/360*πr². So that was the
26:35sector area, and we've already studied
26:38the triangle area—if this side is r,
26:41this side is r, and this is the angle,
26:44I can read it as: 1/2*A*B*sin C, (1/2
26:47AB sin C). So what we can do is, if we
26:50remove the triangle from this entire
26:53sector, then what area will we find? We
26:56will find the area of the minor segment
26:58. Area of the minor segment. Now, what
27:02if I give you this condition in a
27:04question? Well, if the circle is the
27:06same and the examiner tells us in the
27:08exam that they are interested in the
27:10major segment. Suppose the examiner
27:13tells me that this is your A and this
27:15is your B, and you need the area of the
27:18major segment. What is a major segment?
27:21This entire portion, which I am
27:22intentionally highlighting. This entire
27:25portion is your major segment. This
27:28entire portion is your major segment.
27:31The entire upper part is the major
27:32segment. So, what used to we do in
27:34O-Level Maths? We studied this in
27:36O-Level Maths. There are two options to
27:39find the area of the major segment. We
27:42know what this is, right? r, what is
27:43this? r, the radius. What is this? An
27:46isosceles triangle. And what is our
27:49target? Our target is that we need the
27:51angle of the major sector. So, note an
27:53interesting fact: the angle at the
27:55center is always 360, so if the bottom
27:57angle you have is theta, then
27:59definitely, what will this angle be?
28:02This angle will be 360-theta, 360-theta
28:07. Now, if you want the area of the
28:09major segment, let me do one thing, I
28:11will erase this portion and show you.
28:13There you go, which segment is this
28:14whole one? This entire segment is your
28:17major segment. Previously, what were we
28:19talking about? Previously, we talked
28:21about the minor segment, previously we
28:24talked about the minor segment. Minor,
28:27minor, meaning the smaller segment. Now
28:30we are talking about the major segment.
28:32Major, this segment here, this whole
28:34upper portion, what was this segment?
28:38This entire segment is your major
28:40segment. Major segment. Now, if I ask
28:44you that I need its area. Then what
28:47will you do? You will do area of the
28:49triangle. What is the triangle's name?
28:53The triangle's name is area of triangle
28:56AOB, this is your triangle whose center
28:59we can name O, the center's name was O,
29:02area of the triangle plus area of the
29:05major sector, triangle plus major
29:07sector AOB, meaning we will have to do
29:10what? We need to add this triangle and
29:13this entire major sector. If we add the
29:16areas of these two, we will get the
29:18area of the major sector. So, do we
29:20know what the area of the triangle is?
29:21The area of the triangle is 1/2 a b sin
29:24c, what kind of triangle is this? We
29:27call this triangle an isosceles
29:30triangle, and this major sector of
29:32yours, this one, is the entire major
29:35sector. So, how do we get the area of
29:37the major sector? The angle of the
29:39major sector is 360-θ. θ/360, meaning
29:44the angle over the total angle 360*
29:47πr², so if we add both of these areas
29:50, we will find the area of the major
29:54segment. The area of the major segment
29:57will be found. Is that clear, kid? So,
29:59that is how we keep finding the area of
30:01the major segment. Area of the major
30:04segment, or there is one more thing we
30:06can do; if you want a second option to
30:08find its area, another option is
30:10possible. What is the other option if
30:12you want the major area, the area of
30:14the major sector? In math, we can also
30:17calculate the area of the major sector
30:20like this. We can also do one thing:
30:23subtract the area of the minor sector
30:25from the area of the complete circle.
30:29Subtract the area of the minor sector.
30:34That is, our target is to subtract the
30:37area of the minor sector from the
30:39complete area. This is the complete
30:42area. If I subtract this area from it,
30:44then the remaining area will be whose
30:47area? That remaining area will be of
30:50the major sector. So, technically, both
30:52are the same thing. Now, solve it
30:54however you like. You can solve it
30:57using both minor and major styles.
30:59Either solve it by breaking down the
31:01major one. That is, isosceles triangle
31:03plus major sector, or take the complete
31:06circle, whose area is πr², and
31:08subtract the area of the minor one from
31:11it. What will the ultimate result be?
31:13The ultimate result will be exactly the
31:15same. Is that clear, kid? So, we call
31:17this a segment in O-Level Math. Okay,
31:19now we need to look at another
31:21interesting point. Right? I will
31:23explain it in a bit more detail. Look.
31:25Now let's examine the segment in more
31:28detail. Let's examine this same segment
31:30in a little more detail. Look, the
31:32examiner used to test two or three
31:34other things as well. Just like in
31:36O-Level Math, they test it in A-Level
31:38Math too. Look how; suppose I show you
31:41the same sector diagram again. We
31:44studied that this is our chord. And if
31:46we join both points of the chord to the
31:48center, what kind of triangle will be
31:50formed? An isosceles triangle will be
31:53formed. What kind of triangle will be
31:55formed? An isosceles triangle. Alright,
31:57son? Look again. If I join the center
31:59and both points of the chord, what kind
32:02of triangle is formed? Isosceles.
32:05Isosceles. Alright, son? Isosceles. We
32:07have discussed this previously as well.
32:10And let’s name this angle here theta.
32:14So this radius is r. This radius is r.
32:16Okay? Now, the examiner tests two
32:18things. Number one. The examiner says,
32:21suppose you have a chord. There is the
32:23length of the chord. The chord’s name
32:24is AB, that is the chord's name.
32:26Let’s say the chord’s name is AB.
32:28The examiner asks you two or three
32:29questions. Number one, the examiner
32:32will tell you to find the length of the
32:35chord. Find the length of the chord AB.
32:39You have to find this out. The examiner
32:41will tell you to find the length of
32:43this chord. Length of the chord. This
32:46is your chord. Alright, son? This is
32:48your chord and you need to find the
32:50length of this chord. I need the length
32:53of this chord. Length of the chord. So,
32:55in O-Level Math, we used to use
32:57multiple options. Look, I will explain
33:00three or four options to you. Then,
33:02depending on the question, we will
33:04apply the conditions according to the
33:06situation. Option number one. What was
33:09the first option? The first option was
33:11the Cosine Rule. The first option is
33:14that you apply the Cosine Rule. Look,
33:16obviously, this is a length. So, if you
33:19know that this side of the triangle is
33:21r. This side is r and this angle is
33:23theta. Then according to the Cosine
33:26Rule, if you require the length AB, how
33:28will we find it? We applied the Cosine
33:31Rule. So, according to the Cosine Rule,
33:33what will be the answer? a² + b²-2*a*
33:38b*cos theta. We used to call this
33:43formula the Cosine Rule. Cosine Rule,
33:46well, you have another option too. You
33:51know that, obviously, you know that
33:53this angle and this angle here will
33:56both be the same. If I read this angle
34:00as alpha, then this will also be alpha.
34:01Why are both the same? Because the
34:03triangle is what? The triangle is
34:04isosceles. So, if I ask you how to find
34:07alpha? Then you will say 180-theta
34:10divided by 2, meaning what is the sum
34:13of this complete angle? It is 180.
34:16Subtract this angle from 180 and divide
34:18by two, so that these two equal, that
34:20is, identical angles can be found. So,
34:22you can find alpha this way. Now, if I
34:26tell you there is another option, there
34:27is one more way to find the length of
34:29the chord. Just like we used to do in
34:32O-Level math. What is another option?
34:34Another option is that you apply the
34:37sine rule to this same triangle.
34:41Earlier, we applied the cosine rule.
34:43Now, let's apply the sine rule. Look at
34:47the sine rule; your main target is this
34:49side. To find this side. So, we can
34:51also do it this way. If I apply the
34:53sine rule. Then according to the sine
34:55rule, I can say this. According to the
34:57sine rule, I can say this. Chord is AB.
34:59AB/sinθ. Meaning side over angle, and
35:08in the same way, I can also say that r/
35:12sin alpha, r/sin alpha, r/sin alpha;
35:16with the help of this, what will we
35:20find? With the help of this, we will
35:22find the chord. That is, we can also
35:25find the length of the chord by
35:26applying the cosine rule. We studied
35:28this in O-Level math as well. Is that
35:30clear, son? So, one option is the
35:32cosine rule. One option is the sine
35:35rule. There is one more option too. I
35:38will show you the isosceles triangle
35:39again. The triangle you have here is an
35:42isosceles triangle. There, what type of
35:44triangle was this? Isosceles. I have
35:46copied the triangle. The name of the
35:48triangle was OAB. The name of this
35:50triangle was OAB. We know this radius
35:53was R. This radius was R. And what was
35:55this angle? This angle you have was
35:57theta. And this angle and this angle
35:59were both the same because in an
36:01isosceles triangle, these two base
36:02angles are always the same. So, if this
36:05was alpha, then this was also alpha.
36:07Alright? Okay, now if I apply the
36:10property of an isosceles triangle, any
36:12student who knows the property of
36:14isosceles knows that when you drop a
36:16line in an isosceles triangle, it
36:18always bisects the base. Meaning it
36:20will divide it into two right-angled
36:22triangles. And it also divides the
36:24angle equally. θ/2 and θ/2; when you
36:27drop a line in an isosceles triangle,
36:30its job is to bisect the base and also
36:33divide the angle equally. Now you have
36:36two options to find the length of the
36:37chord. Let us consider this small
36:41length as x. Alright? Come, let me
36:44explain it to you like this. If this
36:45small length is x, then obviously this
36:47will also be x. Now you again have a
36:49double option. If you consider this
36:51half right angle. Consider this half
36:54right angle like this. So now you have
36:56two options. This is alpha. This is
36:59your x. Okay? Let's read this point as
37:02the midpoint. So this is your a. This
37:04is your m. This is your o. This is your
37:06r. So now if you apply cos, that is, it
37:09has now broken into a right-angled
37:11triangle. So what will be the answer
37:13for cos? x/r, so if I ask you what x is
37:16equal to? Then you would say R*cos
37:19alpha. So if I ask you how to find the
37:21chord, then you would say what is a
37:23chord? Chord is 2 times of x, chord is
37:262 times of x, here is 2 times of x. So
37:29I can also show it as 2 times R cos
37:32alpha. So there is another style to
37:36find the length of the chord, which is
37:38to break the isosceles triangle, break
37:41the isosceles triangle and find x
37:43through cos, that is, find half the
37:45chord. What is x? x is technically 1/2
37:50chord and then multiply it by two. So
37:53this is the third option to find the
37:56chord length. Third option to find the
37:59chord length. I am telling you this in
38:02such detail because this appears very
38:04frequently in exams. Third option to
38:06find the chord length. Similarly,
38:08option number four to find the chord
38:10length. One more option too. So that
38:13means you have double options. You can
38:16work in any way in the exam. Now
38:18another option could be that you again
38:20consider this same right-angled
38:22triangle. And now you look at this
38:24angle. This angle, the name of this
38:27angle was theta/2, with respect to this
38:29angle, with respect to this angle. This
38:32is opposite. This is hypotenuse.
38:34Opposite and hypotenuse. Clear? We have
38:37applied sin. If I apply sin, then what
38:40is the answer for sin? x/r, x/r, so
38:43from here you have found 1/2 chord.
38:46What is the answer for 1/2 chord? R
38:49times sin theta/2, R times sin theta/2,
38:52this is your 1/2 chord. So your target
38:56is the full chord, meaning AB. AB means
38:592 times X, so just multiply this length
39:01by two, and what’s the answer? 2
39:04times r sin θ/2. So look, I’ve given
39:09you four ways to solve the same
39:11question; one option to find the chord
39:15length is to apply the cosine rule.
39:19What is the second option? You apply
39:21the sine rule. What is the third option
39:23? To break down the isosceles triangle.
39:26You break down the isosceles triangle.
39:29When you break the isosceles triangle,
39:31there is a possibility with respect to
39:32the right angle, and another
39:33possibility with respect to this angle.
39:36With respect to this angle, this is the
39:37base, and this is the hypotenuse. With
39:39respect to this angle, this is the
39:40opposite, and this is the hypotenuse.
39:43This means you have four math options
39:45to find the length of the chord. Which
39:49one is the quickest option? The
39:51quickest option is the cosine rule. In
39:53past papers, it has happened a few
39:55times that the examiner made you use
39:56this approach or that one. That is why
39:59you must first understand the sense of
40:00everything. Clear, kid? So that if
40:03there are no restrictions according to
40:04the situation, we can apply whichever
40:06model we want. The answer will
40:08obviously be the same in the end. Clear
40:09, kid? So what did we discuss? We
40:12discussed the length of the chord. So,
40:15four different math options to find the
40:18chord length. Okay? Okay? If I tell you
40:23the examiner asks us to find the
40:25perimeter of the minor segment, what is
40:27it? Let’s add another condition here:
40:32what if the question in the exam asks
40:36for the perimeter of the minor segment?
40:40The examiner asks you to find the
40:42perimeter of the minor segment.
40:44Perimeter. Then you would say this
40:47length is called the chord length. And
40:49what is this? This is the arc length.
40:51Arc length. So it means, what would the
40:53answer be? Chord length AB, meaning the
40:57length of chord AB plus the arc length
41:01AB. Okay? So what is AB? AB is the
41:04chord length. If I mark it here, what
41:08is AB? AB is the length of the chord.
41:12AB is the chord length. AB is the chord
41:15length. So you will need to add to the
41:18chord length. You will need to add this
41:21arc length to the chord length. Clear,
41:24kid? So, whose perimeter will this be?
41:27This will give the perimeter of the
41:28major-minor segment. And this arc, what
41:31will we call this arc? We will also
41:34read this arc as a minor arc. What will
41:38we read this arc as? We will read this
41:41arc as a minor arc. Minor. Is that
41:44clear, son? So, if you require the
41:46perimeter of the minor segment, what
41:47will you do? Add chord AB and the minor
41:50arc. And if the examiner says that the
41:53perimeter of the major segment is
41:55required. Then what will you do? Add
41:57the major arc length to chord AB. Add
42:00the major arc length. If you need the
42:04perimeter of the minor segment, the
42:05chord and minor arc will be added. If
42:08you need the perimeter of the major
42:10segment, the chord and major arc length
42:12will be added. Is that clear, son?
42:14Meaning, depending on the question, we
42:16solve it according to the situation. Is
42:19that clear, son? So, what was the point
42:21we studied today? This was mensuration.
42:23This is called mensuration of circles.
42:25Meaning, we covered circle, semi-circle
42:28, quadrant, segment, sector, and then
42:31the area of major and minor segments,
42:34and the perimeter of major and minor
42:37segments. In which the length of the
42:40chord played the most important role.
42:42The length of the chord had the most
42:44important role. Is that clear, son?
42:46Finally, the last point of this chapter
42:48. The name of this last point is
42:50radians. Okay, son? After this, we can
42:53finally start this chapter. Okay? The
42:55name of this last point is radians. Now
42:57, let's see what radians are. Okay? If
43:00we remember in O-Level Math, we used to
43:04study a topic. In O-Level Math. The
43:08name of that topic was Measures. In
43:10which we used to look at different
43:12conversions. There is a topic in
43:14O-Level Math named Measures. Meaning,
43:17conversions used to happen in this. We
43:19used to convert data from one unit to
43:21another unit. Is that clear, son? Now,
43:24look how. If any student remembers, we
43:27must have studied different units of
43:29length in O-Level Math. If I talk about
43:31O-Level Math, we must have studied
43:33different units of length in O-Level
43:35Math. In different topics like
43:38mensuration, scales, and maps, the
43:40units of length are meter, centimeter,
43:43kilometer, decimeter, millimeter. What
43:46are all these? Millimeter, these are
43:48all different units. These are units of
43:50what? These are different units of
43:52length. I am starting from O-Level Math
43:54. So, in O-Level Math, we also used to
43:57say that 1 km is equal to. 1 km is
43:59equal to 1000 meters. We used to call
44:021000 'kilo'. So, what did I write here?
44:04I have written a relationship here
44:06between two different units. So if we
44:08define this, we will call it a
44:10relationship. What is this thing? This
44:14is your relationship between two
44:16different units. Meaning, what is the
44:19relationship between two units? I mean,
44:21I know kilometer is a larger unit and
44:23meter is a smaller unit. And we kept
44:26saying 1 meter equals 100 cm, right? So
44:29we call this conversion. So we must
44:32have studied different units of length
44:34in O-Level Math. Similarly, I am
44:35telling you something new now. Today we
44:38are learning something new. Math says
44:40there are different units for angles
44:43too. There are different units for
44:45angles as well. Degree, radian, gradian
44:48. But in A-Level Math, there are only
44:51two units. Okay? In A-Level Math, there
44:54are only two units. Which units are
44:56those? One unit is what you used to
44:58study in O-Level Math. The name of that
45:00unit was degree. And there is a new
45:02unit which was also in O-Level Add Math
45:04. The name of that unit is radians. So
45:07what is a radian basically? A radian is
45:10also a unit for measuring angles. We
45:12use degree very frequently in O-Level
45:15Math. Now we will talk about radians.
45:17So radians is also a unit for measuring
45:20angles. Unit. Clear? One is degree and
45:23one is radians. Look, radians is also
45:26shown in short form like this. When I
45:29explain radian in a little while, we
45:31show radians in short form like this.
45:33RAD, radian. And if anyone remembers,
45:36there used to be this sign above
45:37degrees. Like this, this was the degree
45:40sign. Clear? We kept saying 30 °, 60
45:43°, 40 °, 90 °, 45 °, so we used to
45:47mark a small circle above the degree.
45:50This is the degree sign and radians. So
45:52what is a radian basically? A radian is
45:55a unit of an angle. Well, the unit in
45:58our syllabus for AS Math is radians
46:01only. In this entire topic, we don't
46:03have to use degrees. We have to use
46:05radians. Okay? So just as there is a
46:07relationship between two different
46:08units. Similarly, Math says there is a
46:10relationship between degree and radian
46:12too. There is a relationship between
46:14degree and radians as well. Math says
46:17that 180 ° is equal to pi radians.
46:21This is the relationship between these
46:23two different units. 180 ° is equal to
46:26pi radians. So basically, our required
46:29unit in this entire topic. So, the
46:33topic we are studying, our required
46:36unit, our syllabus, is in the form of
46:39radians. What is our required unit? In
46:43the form of radians. You will hear the
46:45word radians in two or three topics. In
46:47circular measurement, differentiation,
46:50integration, trigonometry, you will
46:52hear the word radians. Radians. Right,
46:55kids? The required unit is radians. So
46:58if a question comes in the exam in
47:00degrees, you will have to convert the
47:03degrees into radians. So your first
47:07task will be to convert the degrees
47:10into radians. Because your syllabus
47:13requirement is radians. Degrees are not
47:16required. Your syllabus itself is in
47:18the radian unit. Meaning, what will be
47:19the unit of the angle? It will be
47:20radians. Calculators also have a radian
47:22feature. So, let’s see how to convert
47:25it. Today we learned that 180 ° is
47:28equal to what? 180 ° is equal to pi
47:30radians. Pi radians. So if I ask you
47:34what 1 ° is equal to? What is 1 °
47:37equal to? Then I can also say that 1 °
47:40is equal to. π/180 radians. π/180
47:47radians. π/180 radians. 1 degree is
47:51equal to. π/180 radians. So if you
47:54want to convert degrees to radians,
47:56what should we do? Multiply by π/180,
47:59see how? Mathematics says that if you
48:03want to convert any degree angle, any
48:07degree angle. Okay? If you want to
48:11convert a degree angle into radians. If
48:13you want to convert any degree angle
48:16into radians. Then what does math say
48:18you should do to that degree angle? You
48:20multiply that degree angle by π/180.
48:25Then your data will be converted from
48:27degrees to what? To radians. Meaning
48:29your data will be in terms of pi. In
48:31terms of pi. How? Let me give you a
48:34small example. Suppose I want to
48:38convert a 60 ° angle into radians. I
48:41want to convert a 60 ° angle in terms
48:45of radians. Meaning I have to convert
48:48the 60 ° unit into radians. This is to
48:49convert into radians. Clear? Alright.
48:52So what is option number one? I just
48:54told you 2 minutes ago that if you want
48:57to convert degrees to radians, what
48:59will you do? The first option is that
49:01if you require the angle in radians,
49:03what will you do? Multiply 60 ° by π/
49:08180. Multiply 60 ° by π/180. Clear,
49:14son? So now what happens? This 60 times
49:173 is 180. So your data is converted
49:19into what? Your data has converted into
49:21radians. So now what has it converted
49:23into? It converted into π/3. So that
49:25means 60 ° means π/3, if anyone has a
49:27doubt, you check it. 180 means pi. So
49:30it converted back into degrees as 60 °
49:33. Okay? So what does pi radians mean?
49:36180 °. The second option is this. The
49:39second option is that you use the basic
49:42unitary method. By the way, both these
49:44methods are the same. You use the
49:46unitary method or the ratio method.
49:49Okay? You use the unitary method, which
49:51we also call the ratio method in simple
49:53language. Like we use very frequently
49:55in O-level math. Look, here is degrees
49:57and here is radians. We know that 180
50:00° is equal to pi radians. I need to
50:03convert what 60 ° is. So I consider it
50:07x radians. So what did we do? Like in
50:09O-level math, we cross-multiplied. That
50:12is why this is called the unitary
50:14method or ratio method. We
50:15cross-multiplied. So when we
50:17cross-multiplied, what did the answer
50:19for x become? 60*π/180, so indirectly
50:22this is the same unitary method. So
50:25what will the answer for x be derived
50:27as? π/3, meaning the same answer is
50:31derived. Clear, son? So what should you
50:34do? If you want to convert degrees to
50:36radians, multiply those degrees by π/
50:39180. Otherwise, apply the unitary
50:41method or the ratio method. You see,
50:43ultimately what will the answer for
50:45both these methods derive? Finally,
50:47both these methods derive the same
50:48answer. Okay? Finally, what will the
50:50answer for both these methods derive?
50:52The same. Okay? See, I have copied both
50:56options. Whichever method you find easy
50:58, you can apply that method. What will
51:00the result from both be? The result
51:02from both will obviously be the same.
51:04Clear, son? Both options are perfect.
51:07Meaning now it depends on the child, if
51:09a student multiplies the degrees by π/
51:10180, the result will still be the same.
51:13Otherwise, apply the unitary method,
51:14then too the result will be what? The
51:16same. So basically, what is required?
51:19What is our target? Our target is
51:21radians itself. So let me show you how
51:23to convert some frequent values into
51:26radians. Frequent values. Here I have
51:29copied degrees on one side. Okay? I
51:32have copied the degrees on one side
51:33here. Meaning, I am showing you how to
51:36convert degrees into radians. I have
51:38copied the radians here. Degrees and
51:41radians are both units of angles. And
51:44here I will show you the process of
51:46conversion. Working, suppose I want to
51:50find out what 30 ° is. I need to
51:52convert 30 ° into radians. So what
51:54will I do? I will multiply 30 ° by π/
51:58180. So what answer will be derived
52:00from here? π/6, that's it. Done. So
52:03it's a matter of seconds. It gets
52:05solved in seconds. This 30 ° has been
52:07converted into radians. If I ask you, I
52:10need to convert 45 °. I don't know
52:12what 45 ° is. So what will I do? If I
52:15multiply 45 ° by π/180, then 45 times
52:194 is it. So what answer was derived
52:21from this? The answer derived was π/4;
52:24what was the answer derived? The answer
52:27derived was π/4, or if I ask you that
52:29I want to convert 60 ° into radians. I
52:33need to convert 60 °. So how will 60
52:35° be converted? 60*π/180, 60*π/180,
52:39so what answer was derived from this?
52:44The answer derived was π/3, okay? So
52:48these are some frequent values that you
52:50will see a lot in this topic. If I ask
52:53you to find out what 90 ° is. I don't
52:55know what 90 ° is. So what will I do?
52:58If I multiply 90 ° by π/180, then 90
53:02times 2 is it. So what answer was
53:04derived from here? The answer derived
53:07from here was π/2, the answer derived
53:10from here was π/2, π/2, this is the
53:14answer derived. Similarly, if I ask you
53:18that I need to find out what 270 ° is.
53:22Let's suppose I need to find out 270 °
53:24, so what will I do? I will multiply
53:27270 by π/180. 0 and 0 cancel, 9*3, 9*2
53:31, so what answer was derived? The
53:34answer derived was 3 times π/2, 3
53:37times π/2, clear dear? So what was
53:40ultimately derived? The same was
53:42derived. So what did I do? Whichever
53:45angle needed to be converted, I
53:47multiplied it by what? I multiplied it
53:50by π/180. So the angle that was in
53:52degrees has been converted into radians
53:54. Similarly, if I ask you, man, I need
53:58to find out what 360 ° is? So what did
54:00I do? I multiplied 360 ° by π/180. So
54:05what result was derived? 2π was the
54:07result derived. Anyway, it's common
54:10sense. If 180 is pi, then 360 is double
54:13of 180, so it becomes 2π. So what did
54:16we just do? I showed you different
54:19examples of converting degrees to
54:21radians. Suppose I want to find out
54:25what 135 ° is? Just a random example,
54:28so what did I do? I multiplied 135 by
54:31π/180. So 45*3 = 45*4, what answer was
54:36derived? 3π/4, so in all these
54:39examples, I showed you the conversion
54:42of different degrees into radians. So
54:45what is 30 °? π/6, so whenever you
54:48want to convert degrees to radians,
54:51just multiply that degree by π/180. So
54:5530 ° means π/6, 45 ° means π/4,
54:58similarly 60 ° means π/3, 90 ° means
55:02π/2, so what did I do for you? I
55:06showed you how to convert degrees to
55:08radians. And if you have to convert
55:10radians back, then it's very easy. Just
55:12put 180 in place of pi. It converted
55:14back. Put 180 in place of pi. It
55:18converted back to 45. Put 180 in place
55:21of pi. It converted back to 60. Put 180
55:24in place of pi. It converted back to 90
55:27°. Clear, kid? Put 180 in place of pi.
55:30It converted back to 360 °. This is a
55:33matter of seconds. So basically, what
55:35are we interested in? If your angle is
55:37given in degrees in the question, then
55:39you convert that given question's angle
55:42from degrees into what? Convert it into
55:44radians. Clear, kid? There are a couple
55:47more small points. Then finally we will
55:49start circular measurement. Okay? Look
55:52at one more example. Today we studied a
55:55topic. The name of the topic was sector
55:57. So let's have you look at one more
55:59thing again. We studied a point today
56:01as well. The name was sector, sector.
56:03Right? Just a quick recap. I'll show
56:05you the sector again. We studied this
56:07today. We studied this today, what is a
56:10sector? A sector is the area bounded
56:12between two radii and an arc length.
56:15Area bounded between two radii and an
56:18arc length. We call this a sector. We
56:20studied this today, today. So, if this
56:22is a minor sector. We learned today
56:25that if this is a minor sector, then
56:26what will the angle below be? The angle
56:29below will be the major one. What will
56:30the angle below be? Major, so what will
56:32the angle below become? 360-theta. 360-
56:37theta. That’s today's lecture, by the
56:39way. Okay? Alright, now there are two
56:40issues. We have to check this first.
56:42Look, previously, if I talk to you
56:45about the minor sector, this is minor.
56:47Minor. Well, I told you today, I told
56:50you today, let me repeat it again, that
56:53whenever you need to convert degrees. I
56:56told you today that whenever you need
56:58to convert degrees into radians.
57:00Meaning you are converting one unit of
57:02an angle into a second unit. If you
57:05convert degrees to radians, what will
57:07you do? Look, I told you that if you
57:10want to convert degrees to radians, you
57:13multiply that degree angle by pi over
57:16180. I explained this same thing to you
57:18on the previous page. So your data will
57:20convert from degrees to what? Radians.
57:22So if I tell you what is 360 °? Then
57:25how will you convert 360 °? I just
57:27showed you how to do it. Multiply 360
57:30by pi over 180. Clear, son? So that
57:34means from now on, whenever the word
57:38360 is used, 360 means 2π radians, by
57:41the way, 2π radians. So what do we
57:45call 360 ° in radians? In radians, we
57:49call it 2π, 2π, clear son? We call
57:52360 ° in radians as 2π radians. Clear
57:55? Well, if you remember today's lecture
57:57, today's. So today we studied a sector
58:00, minor. What did I do? I copied the
58:02minor sector separately. Okay? I copied
58:05this minor sector separately. Minor.
58:07Now you see, what is this angle of the
58:09minor sector? The angle of the minor is
58:11, let's say, theta. Here you go, the
58:12angle of the minor is theta. This is
58:14the radius. This radius. So today you
58:16studied a formula, today. Today you
58:19learned what the formula for the arc
58:20length of the minor sector is. Theta/
58:23360*2πr, this is the formula you
58:26studied today. The issue is that this
58:30formula is in degrees and my syllabus
58:32is in radians. So finally, what new
58:34thing will I do today? The new thing
58:37could be that I replace 360 with 2π.
58:41So when 2π and 2π cancel out, today I
58:43discovered a new thing: what is the new
58:46formula for arc length? It's r*theta.
58:50What does s show in this? s is arc
58:53length. Which we used to study before
58:56this. What is s? s is arc length.
58:59Length of arc or arc length. What is r
59:02in this? r used to be radius. Okay? Let
59:06me explain this in more detail. What is
59:08the new formula we studied today? s = r
59:10theta. Okay? So if I write it in more
59:14detail, what was s? We know what s is,
59:16right? In the language of math, we read
59:20s as arc length. s means arc length. If
59:23I ask you what r is? Then we will say r
59:26means radius. If I ask what theta is?
59:30Then we will say, which angle is theta?
59:32Theta is the angle in radians. Angle in
59:36radians. Clear, kid? So now, your old
59:39formula which was in terms of degrees
59:41has converted from degrees to what
59:43today? Converted to radians. So from
59:46today onwards, we will use the formula
59:49in radian form. Clear, kid? Radian form
59:52. That's why I showed you how to
59:54convert the formula first. Okay? And
59:56this formula is also given in A-Level
59:59Maths. The formula table provided in
1:00:01A-Level Maths is called the MF 19 table
1:00:03. So this formula is given in
1:00:06Mathematical Formula Table 19. Clear,
1:00:08kid? It will be given in the paper.
1:00:10Well, then we saw one more thing today.
1:00:12We studied that this is also a
1:00:14carry-forward from O-Level Maths. What
1:00:16was the formula for area? Wait, let's
1:00:18see. Now let's look at the area formula
1:00:20. So what did we study before regarding
1:00:21the area formula? θ/360*πr², we
1:00:25studied this formula previously. But
1:00:31the issue is that now I don't want to
1:00:33use the angle in degrees. I don't have
1:00:35to use the degree angle. So what will I
1:00:38do? θ/2π*πr², so now look, π and
1:00:42π have done what? They canceled out.
1:00:45So from here, I got a new formula for
1:00:47area. So what is the new formula for
1:00:49area? 1/2 r² theta, this formula will
1:00:53be used from today for the area of a
1:00:57sector. This formula will be used from
1:01:00today for the area of the sector.
1:01:03Similarly, look at one more new thing.
1:01:05We just derived that the area of a
1:01:07sector formula is 1/2 r² theta. Just a
1:01:10minute ago we saw that this area
1:01:12formula is 1/2 r² theta. I can also
1:01:14break this formula down like this. 1/2*
1:01:17r*r, because what is r*r? r squared. So
1:01:21now I know one more thing, what is r*
1:01:24theta called? r*theta is also called s
1:01:27in math. r*theta is also called arc
1:01:30length in math. So this means a new
1:01:32formula is derived, 1/2 rs. So a new
1:01:37formula for the area of a sector is
1:01:39derived. The new formula derived for
1:01:42the area of a sector is 1/2 RS, which
1:01:44means there are two new formulas in
1:01:47A-Level Math. By the way, both these
1:01:49formulas are the same. Both these
1:01:51formulas are given in MF19. Meaning,
1:01:54this formula will be given in the exam
1:01:56and this formula will also be given in
1:01:57the exam. So both of these formulas are
1:02:01given in MF19. What is R in these two
1:02:04formulas? R is for radius. R for radius
1:02:07. What is s in these two formulas? We
1:02:10used to call s the arc length. What is
1:02:14theta in this? Theta is called the
1:02:17angle in radians in math language.
1:02:20Angle in radians. So these are the two
1:02:22new formulas. So from today onwards,
1:02:26whenever we use the formula, we will
1:02:29always use it in radians form. Clear,
1:02:32kid? We will use it in radians form.
1:02:36Okay? Finally, one last thing, we
1:02:39should know this option on the
1:02:41calculator. There is a radians option
1:02:43on the calculator. Okay? There is an
1:02:46option for radians on the calculator.
1:02:47Okay? Let me tell you one more small
1:02:49thing. One more small thing. Then
1:02:52finally we will start our topic on
1:02:54circular measure. Well, you need exact
1:02:57values in the exam. Many such questions
1:03:00will come. You will frequently hear a
1:03:02word. That word is exact values. Very
1:03:05frequently. The examiner will ask you
1:03:08to find the exact value of the area.
1:03:10Find the exact value of the perimeter.
1:03:13So, this word "exact value" means an
1:03:16answer without decimals. Without
1:03:21decimal answer. Without decimal values.
1:03:25Without decimal answers. Meaning,
1:03:28whenever this word appears in the exam,
1:03:30if you are required to find the area or
1:03:33perimeter, then the exact value of the
1:03:35area or perimeter means your answer
1:03:37should be without decimals. You will
1:03:40show the question in that same form.
1:03:42Whether there is pi in the question, or
1:03:44there is a surd in the question.
1:03:45Whether there is a surd in the question
1:03:47. The very first task in this entire
1:03:50topic is to always convert your
1:03:53calculator to radians. The calculator's
1:03:56angle unit must always be in radians.
1:03:59The most important thing is that the
1:04:01calculator's angle unit should always
1:04:03be radians. Okay? You must always
1:04:05convert the calculator to radians. Your
1:04:07calculator should always be in radians.
1:04:10Now, there are some values that your
1:04:12calculator automatically shows in exact
1:04:15form itself. How? If I tell you to use
1:04:18your calculator. Put it into the
1:04:20calculator. Put in sin π/4. When you
1:04:25input it into the calculator, since π/
1:04:274 means 45, in what form will the
1:04:29calculator automatically show the
1:04:31answer? The calculator will show the
1:04:34answer as √2/2; the calculator
1:04:36already has this feature. I am not
1:04:38doing anything here. The calculator
1:04:40already has this feature. Meaning the
1:04:42calculator will automatically show the
1:04:43answer in surd form. Surd. By the way,
1:04:47if anyone remembers, π/4 means 180/4,
1:04:49which already means 45 °. So, in
1:04:53degrees we used to say sin 45, in
1:04:56radians we will say sin π/4; the
1:04:58calculator's angle unit must be in
1:05:01radians. Is that clear, son? The
1:05:02calculator will do this task itself. If
1:05:05I tell you, what answer will the
1:05:07calculator show for cos π/6? What
1:05:10answer will it show for cos π/6? This
1:05:12feature is already inside every
1:05:15calculator. So, when you put it into
1:05:17the calculator, what answer will it
1:05:19automatically show? √3/2, this works
1:05:23because every calculator has a surd
1:05:26feature. Perhaps we remember that surds
1:05:29are those roots whose answer is derived
1:05:31in decimals or whose answer is a
1:05:33decimal. Is that clear, son? So, this
1:05:34is a built-in feature inside the
1:05:36calculator. If you press the decimal
1:05:37button or the equals button on the
1:05:39calculator, it will show the same
1:05:40answer in decimals. For example, for
1:05:43√3/2, your calculator will show 0.866
1:05:45. This task works because the
1:05:47calculator has both features. The
1:05:49calculators that are coming out these
1:05:51days are called the Natural Display
1:05:53series. Every calculator, every
1:05:56calculator's name is Natural Display
1:05:59series. Natural Display series. All the
1:06:02new calculators are named Natural
1:06:04Display series. They already have this
1:06:07surd feature. Meaning we are not doing
1:06:09anything extraordinary. The calculator
1:06:11will already have the surd feature.
1:06:14Okay? The calculator will already have
1:06:16the surd feature. I am just telling you
1:06:19some particular values that your
1:06:21calculator will show itself. We are not
1:06:24memorizing anything. The calculator
1:06:25already has this feature. So, when you
1:06:27input the angle into the calculator, it
1:06:28will show the answer itself. Is that
1:06:30clear, son? So, I am showing you the
1:06:32exact form. Look at one more thing. If
1:06:34you input into the calculator, what is
1:06:37the answer for tan π/6? There are some
1:06:40particular values that are used very
1:06:42frequently. tan π/6. Now, π/6 means
1:06:45180/6, which is 30 °. So, what will
1:06:48your calculator show? Your calculator
1:06:52will show the answer for tan π/6 as
1:06:54√3/3; this feature is already in the
1:06:57calculator. It is already a feature in
1:06:59the calculator. So, what will the
1:07:01calculator show? If you press the
1:07:02decimal button, the calculator will
1:07:04show .577. This feature is already in
1:07:07the calculator. Similarly, if I give
1:07:09you an example, what will be the answer
1:07:11to tan π/3? By the way, what does π/3
1:07:13mean? 180/3, which is 60 °. So, what
1:07:16will your calculator automatically show
1:07:20? Your calculator will show √3, and
1:07:23the answer to √3 is 1.732. Is that
1:07:28clear, son? So, all these values are
1:07:30already in the calculator. Meaning, we
1:07:32are not memorizing them by rote. We
1:07:33have also studied these values in
1:07:35O-Level Math. But in O-Level Math, we
1:07:36must have studied them in degrees.
1:07:39180/4 is 45 °, 180/6 is 30 °, 180/6
1:07:41is 30 °, 180/3 is 60 °. These are the
1:07:45values that are frequently in the
1:07:47calculator. Similarly, if you put cos
1:07:52π/3 into the calculator, the
1:07:54calculator shows half. The calculator
1:07:57shows half. So, all these values are
1:07:59already in the calculator. But when you
1:08:02read the questions, the examiner will
1:08:04ask you for exact values in many of
1:08:06them. So, whenever the examiner uses
1:08:10the term "exact values," it means that
1:08:13you must always show the answer in surd
1:08:15form. Meaning, you should never show
1:08:19the answer in decimal form. Just as
1:08:22there is a degree option in the
1:08:23calculator, the next option is radians.
1:08:26So, you will set the calculator to
1:08:28radians. You will convert it. The rest
1:08:30of the work is for the calculator. So
1:08:32today, our basics for this entire topic
1:08:34are finally complete. Is that okay, son
1:08:36? Now, God willing, from here on, we
1:08:39will convert the calculator to radians
1:08:41and practice this topic further. Is
1:08:44that okay, son?

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