Pre-Calculus
Trig Unit 3 Review
Name: _____________________________________
For numbers 1 – 8, use the following formulas to simplify the expression or verify the identity.
Reciprocal Identities
Even-Odd Identities
Pythagorean Identities
sin(–x) = –sin x
cos(–x) = cos x
tan(–x) = –tan x
sin2 x + cos2 x = 1
csc(–x) = –sin x
sec(–x) = sec x
cot(–x) = –cot x
1 + cot2 x = csc2 x
1 + tan2 x = sec2 x
sec 2 t
tan t
1. tan(–x) cos x
2. cos θ csc θ
3.
4. cot(–x) sin x
5. cot(–x) sin (–x)
6. tan θ csc θ cos θ
7. cot x sec x sin x
8. sec α – sec α sin2 α
Sum Formulas
Difference Formulas
cos (α + β) = cos α cos β – sin α sin β
cos (α – β) = cos α cos β + sin α sin β
sin (α + β) = sin α cos β + cos α sin β
sin (α – β) = sin α cos β – cos α sin β
tan (α + β) =
tan (α – β) =
For numbers 9 – 14, use a sum or difference formula to find the exact value of the expression.
9. cos (120° – 45°)
10. sin (60° – 45°)
11. tan (30° + 45°)
 3π π 
+ 
12. cos 
 4 6
 5π 
13. sin 

 12 
 5π π 
+ 
14. tan 
 3 4
For numbers 15 – 17, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.
7π
π
7π
π
cos − cos
sin
15. sin
12
12
12
12



16. sin 40 cos 20 + cos 40 sin 20

4π
5
17.
π
4π
1 − tan tan
5
5
tan
π
5
+ tan
For numbers 18 – 21, use any of the previous formulas to simplify each expression.
π

18. cos  x − 
2

19. cos (α + β ) + cos (α − β )
π

20. tan  θ + 
4

3π 

21. sin  x +

2 

22. Use sin α =
4
7
, α lies in quadrant I, and sin β = , β lies in quadrant II to find the exact value of the following under the given
5
25
conditions.
a) cos (α + β)
b) sin (α + β)
c) tan (α + β)
Double-Angle Formulas
Three Forms of the Double-Angle Formulas for cos 2θ
For numbers 23 and 24, use the given information to find the exact value of each of the following.
a) sin 2θ
23. sin θ =
b) cos 2θ
12
, θ lies in quadrant II.
13
c) tan 2θ
24. cos θ =
40
, θ lies in quadrant IV.
41
For numbers 25 – 27, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the
expression.
2 tan
25. 2 sin 22.5° cos 22.5°
2
2
26. cos 105° – sin 105°
27.
π
8
1 − tan 2
Half-Angle Formulas
For numbers 28 – 30, use a half-angle formula to find the exact value of each expression.
28. sin 15°
29. cos 157.5°
30. tan
3π
8
π
8
31. Use tan
=
α
a) sin
α
2
8
, 180 < α < 270 to find the exact value of each of the following
15
b) cos
α
2
c) tan
α
2