Saturday X-tra
X-Sheet: 12
Trigonometry
Compound Angles and Transformation
Key Concepts
In this session we will focus on summarising what you need to know about:
 Simplifying using compound angles
 Identities
 Solving equations
 When an identity is valid
Laws / Key Concepts / Diagram
cos(   )  cos cos   sin  sin 
cos(   )  cos cos   sin  sin 
sin(   )  sin  cos   cos sin 
sin(   )  sin  cos   cos sin 
sin 2  2 sin  cos 
cos 2  cos2   sin 2 
cos 2  2 cos2   1
cos 2  1  2 sin 2 
cos  b
b  [1 ; 1]
   cos b  k 3600 k  Z
sin   b
b  [1;1 ]
1
  sin 1 b  k 3600
tan   b b  
  tan 1 b  k 3600
or
  1800  sin 1 b  k 3600 k  Z
k Z
X-ample Questions
1. If cos 200  m , write each of the following in terms of m:
a) cos 3400
b) sin110
0
c) 1  sin 70
2
d)
cos 40
0
0
e) cos 50
0
Page 1
2.
Simplify without using a calculator
sin 50 cos15 sin15
cos140
3.
Prove the identity


cos A [cos   A   sin 180  A tan A]  cos 2 A
0
4.
Prove the following identity: tan  1  cos 2
5.
a) Prove that sin( A  B)  sin( A  B)  2 cos A sin B
sin 2
b) Hence, or otherwise, prove that sin 3 x  sin x  2 cos 2 x sin x
6.
Prove that tan x  1  cos 2 x , and hence deduce that
sin 2 x
tan 22
1
o

2 1
2
7.
Determine the general solution
a)
cos 540 cos x  sin 540 sin x  sin 2 x
b)
4 sin x  3cos x  0
c)
sin( x  300 )  cos 2 x  0
d)
2 sin( x  30 0 )  3 cos x
e)
8.
3 cos   sin 
For which values of A is the following identity not valid:
2 sin 2 A
1

2 tan A  sin 2 A tan A
Page 2
X-ercise
1.
2.
cos45   x . cos45   x   12 cos 2 x.
a)
Prove that
b)
Find the general solution if
1
2
sin 2 x cos 40  cos 2 x cos310
Prove the identity:
a)
b)
3.
For what values of
is the identity undefined?
Evaluate and simplify:
sin 155. cos 25
cos320
4.
a)
It is given that tan 22 = p.
Now, express in terms of p, each of the following:
5.
i)
tan 338
ii)
sin 68
iii)
sin 44
Simplify without the use of a calculator:
cos330.sin140
(1)
sin  160  .tan 405.sin 290
(2)
6.
7.
sin 140  cos150 
sin 110  sin 340 
sin x. sin 2 x
 cos 2 x  1
cos x
sin   cos(  30)
Determine the general solution:
Prove the following identity:
Page 3
Answers
1b) x  5 0  k180 0 or x  55 0  k180 0 k  Z
2b) x  90 0  k 360 0 or x  120 0  k 360 0 k  Z
1
3.
2
4.
(1)  p
1
( 2)
1 p2
(3)
2p
1 p2
5. (1)  3
(2)  3
  60 0  k180 0
7.
k Z
Page 4

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