Trigonometry | SOH CAH TOA | Sin, Cos, Tan — Transcript

Full transcript

1. 0:07in today's video we're going to look at
2. 0:09how we can find unknown angles or sides
3. 0:12in right angled
4. 0:13triangles the first thing you need to
5. 0:15know is that you can normally tell when
6. 0:17a triangle is a right angled triangle
7. 0:20because it will have this little square
8. 0:22in one of the corners which tells us
9. 0:25that it's 90° so a right
10. 0:27angle the second thing to know is that
11. 0:31this side opposite the right angle is
12. 0:33always called the hypotenuse which we
13. 0:36represent with the letter H and this
14. 0:38will always be the longest of the three
15. 0:41sides now how we label the other two
16. 0:44sides changes depending on which angle
17. 0:47we're interested in
18. 0:48finding for example if we were
19. 0:50interested in finding this angle in the
20. 0:53bottom right which would label as X then
21. 0:57this side which is opposite X would be
22. 1:01called the opposite which we represent
23. 1:03with the letter O and then this side
24. 1:06next to our X angle or in other words
25. 1:10adjacent to it would be called the
26. 1:12adjacent so we label it with the letter
27. 1:16A on the other hand if we had this other
28. 1:19triangle and we were interested in
29. 1:21finding this top angle instead then the
30. 1:25longest side will still be the
31. 1:27hypotenuse as always but there bottom
32. 1:30side would be the opposite o because
33. 1:33it's opposite the angle we're trying to
34. 1:34find and then this left side would be
35. 1:37the
36. 1:38adjacent now to help us work out the
37. 1:41unknown angles and sides we're going to
38. 1:44be using three different
39. 1:46equations sin x equals opposite over
40. 1:50hypotenuse cos or cosine x equals
41. 1:54adjacent over
42. 1:56hypotenuse and tan equals opposite over
43. 2:00adjacent unfortunately you have to
44. 2:02remember all of these off by heart but
45. 2:06what's easier is just to remember the
46. 2:08phrase
47. 2:09so TOA and then you can work out the
48. 2:13equations from that for example if we
49. 2:16look at saw the S stands for
50. 2:20sinx the O stands for opposite and the H
51. 2:24stands for
52. 2:26hypotenuse so we know that sin x equals
53. 2:29opposite over
54. 2:31hypotenuse meanwhile for C the C stands
55. 2:36for cos or cosine X the a stands for
56. 2:41adjacent and H stands for
57. 2:43hypotenuse and the same rules apply to
58. 2:46TOA t for Tan o for opposite and a for
59. 2:51adjacent if it helps you can write
60. 2:53Soaker TOA as s = o over h c = a over h
61. 3:00and T = O A which makes it a bit easier
62. 3:04to spot the
63. 3:07equations to see how these equations
64. 3:10work in practice though let's have a
65. 3:12look at this question where we've got to
66. 3:15work out the value of the unknown angle
67. 3:19X now whenever you get a question like
68. 3:21this the first thing that you want to do
69. 3:24is label all of the sides so this one is
70. 3:28the hypotenuse because it's the longest
71. 3:30side and opposite the right
72. 3:33angle the bottom one is the adjacent
73. 3:36because it's next to our angle that
74. 3:38we're trying to find and this one is the
75. 3:41opposite because it's opposite our angle
76. 3:44X the next thing to do is note which
77. 3:47sides you know the length of and in this
78. 3:50case we know the length of the opposite
79. 3:53which is
80. 3:5315 and the adjacent which is 11 but we
81. 3:58don't know how long they have
82. 4:00uses and we don't need to find out
83. 4:03either so next we need to look at the
84. 4:06three
85. 4:07equations and find which one we can use
86. 4:10with the information that we've been
87. 4:11given if we start with sin x equals
88. 4:15opposite over
89. 4:16hypotenuse we can't use this one because
90. 4:19we don't know the length of the
91. 4:21hypotenuse and if you look at cosx
92. 4:24equals adjacent over hypotenuse we've
93. 4:26got the same problem again this leaves
94. 4:29us with the Tan x equals opposite over
95. 4:32adjacent equation then which we can use
96. 4:36because we do know what the opposite and
97. 4:38the adjacent lengths
98. 4:40are so if we rewrite this out plugging
99. 4:44in the values that we know we get tan x
100. 4:47= 15 the opposite / 11 the
101. 4:53adjacent now solving this is actually a
102. 4:56bit weird because s cos and tan are all
103. 5:00functions rather than numbers and this
104. 5:03means that we can't just divide both
105. 5:06sides by tan to get X by
106. 5:08itself instead we have to use the
107. 5:11inverse tan function which is tan to
108. 5:14the^ of minus one and on your calculator
109. 5:17you can probably find it by pressing the
110. 5:19shift button and then pressing the tan
111. 5:22button when we apply this to both sides
112. 5:25we end up with X by itself on the left
113. 5:29and T -1 of whatever we had on the right
114. 5:32so in our case
115. 5:3410-1 of 15 11 and then all you have to
116. 5:39do is put that into your
117. 5:41calculator and you should get
118. 5:4453.7 and remember because we're working
119. 5:46out the angle of X it's 53.7
120. 5:51de and that's it that's our
121. 5:56answer let's have a go at one more
122. 6:00so in this question they're asking us to
123. 6:02calculate the length of
124. 6:05PQ which is this length here so we can
125. 6:09go ahead and put a question mark on
126. 6:11there because that's the unknown one
127. 6:12that we're trying to find then like
128. 6:15before we're going to want to label our
129. 6:17triangle so this side we're looking for
130. 6:20is the
131. 6:21hypotenuse this bottom side of 20 cm is
132. 6:25the
133. 6:26adjacent this side on the right is the
134. 6:28opposite and this angle of 35° is our X
135. 6:33this time it's the opposite that we
136. 6:35don't really care about because we
137. 6:37haven't been given its value and we're
138. 6:39not trying to find it so we need to find
139. 6:42the equation that has the adjacent and
140. 6:44the hypotenuse in it which will be cos
141. 6:47xal adjacent over
142. 6:50hypotenuse because we're looking for the
143. 6:52hypotenuse though we're going to have to
144. 6:54rearrange the equation to get the
145. 6:55hypotenuse by itself which we can do by
146. 6:59multiplying both sides by the
147. 7:01hypotenuse and then dividing both sides
148. 7:04by cos x to get hypotenuse equals
149. 7:07adjacent over cos
150. 7:09x then the last thing we need to do is
151. 7:12plug the numbers in so H the hypotenuse
152. 7:16is 20 over cos
153. 7:2035 which if you put it into your
154. 7:22calculator should give you 24
155. 7:25cm one last thing to mention is that if
156. 7:28you ever get a calculator error in this
157. 7:29sort of question it could be because you
158. 7:31didn't close the bracket after the angle
159. 7:34so just remember that after writing C 35
160. 7:37you have to put this closing
161. 7:43bracket if you haven't heard yet you can
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