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Math 12 Pre-calculus
Double/Half angles of trig functions
4
3
and cos K  then find the exact values of sin(2 K ) and cos(2 K )
5
5
1
2. Given that sin( m2 )   .
2
1.
If sin K 
Where are the possible locations of angle m? Is it possible to definitively determine the value of sin( m) ?
a)
cos( m2 )  0 . Find the exact value of tan(m)
7
5. If  is acute and cos(2 )   , find the exact values of cos( ) and sin( )
9
b) It is now known that
6. Use an appropriate formula to re-write as a single trig function
a) 4cos(x)sin(x)
b) 2sin2(x) – 1
c) 1 – 2cos2(3x)
d) cos ( 2x )  sin ( 2x )
2
2
7. Prove each identity:
a)
b) cos x  sin x  cos 2 x
(sin x  cos x)2  1  sin(2 x)
4
9. Solve on the interval 0  x  2
a) sin(2x) + sin(x) = 0
4
b) cos(2x) + 3cos(x) = 1
10. Evaluate exactly
 

 16 
 

8
a) sin 
11. If sin     
b) sin 
3
and  is a 4th quadrant angle, find the exact value of (be careful with signs)
5
 
b) sin  
a)

cos
2

2
12. Consider the diagram below to be accurate; e.g. angle A appears to be between
and

2
A

4
B
(450 and 900). Assuming that each angle is less than one full rotation, find the
sign (+/-) of
a)
c)
e)
g)
i)
sin(2A)
tan(2C)
sin(A+B)
tan(A-C)
sin(
D
2
b) cos(2B)
d) sin(2D)
f) cos(D-C)
h) cos( C2 )
D
C
j) tan( C2 )
)
Answers for most….
24
25
 257
1/9 root(3)
0, 23 ,  , 43
1
4
15
4
10
10

3
,  , 53
 31010
 4 95
 18
 3 87
2 2
2
(why two answers?)
1
3
2 2
3
2sin(2 x)
2  2  2 (you needed to find
2
 cos(2 x)  cos(6 x) cos( x)
cos  8  , which is
2 2
2
)
Math 12 Pre-calculus
Trigonometric Identities
Primary Identities
1
 csc x
sin x
1
 sec x
cos x
1
 cot x
tan x
Quotient Identities
tan x 
sin x
cos x
cot x 
cos x
sin x
Pythagorean Identities
sin 2 x  cos 2 x  1
1  cot 2 x  csc 2 x
tan 2 x  1  sec 2 x
Even / Odd Identities: Sin( ) and Tan( ) are odd, Cos( ) is even
sin( x)   sin x
cos( x)  cos x
tan( x)   tan x
Compound Angle Identities:
sin( A  B)  sin A cos B  sin B cos A
sin( A  B)  sin A cos B  sin B cos A
cos( A  B)  cos A cos B  sin A sin B
cos( A  B)  cos A cos B  sin A sin B
tan( A  B) 
tan A  tan B
1  tan A tan B
tan( A  B) 
tan A  tan B
1  tan A tan B
Double and Half Angle Identities:
cos(2 x)  cos 2 x  sin 2 x
or
sin(2 x)  2sin x cos x
cos(2 x)  2 cos 2 x  1
tan(2 x) 
or
2 tan x
1  tan 2 x
cos(2 x)  1  2sin 2 x
With the sine and cosine half-angle formulae, you will need to draw a picture of the original angle (x) and where
the half-angle (x/2) is, and use it to decide if you want the + or – root.
sin  2x   
1  cos x
2
cos  2x   
1  cos x
2
tan  2x  
1  cos x
sin x