Advanced Math
Trigonometric Identities 5.4, 6.1
Sums & Differences 6.2
REVIEW
Name
March 2015
Use Trigonometric Identities to write each expression in terms of a single trigonometric identity or a constant.
1. cot 𝜃 sin 𝜃
2.
1 − sin2 𝜃
cos2 𝜃
1.)_________________
2.)_________________
3.
1 − sin2 𝜃
sin2 𝜃
4. sin 𝜃 sec 𝜃
3.)_________________
4.)_________________
5.
cos 𝜃
sec 𝜃
6. csc 𝜃 sin 𝜃
5.)_________________
6.)_________________
1
Use Trigonometric Identities to write each expression in terms of a single trigonometric identity or a constant.
7.
tan 𝜃 cos 𝜃
sin 𝜃
8.
tan 𝜃
7.)_________________
cot 𝜃
8.)_________________
9. cos 𝜃 csc 𝜃 tan 𝜃
10. cos 𝜃 csc 𝜃 cot 𝜃
9.)_________________
10.)________________
11.
sin2 𝜃
1 + cos 𝜃
12. cos 𝜃 +
sin2 𝜃
cos 𝜃
11.)________________
12.)________________
13. tan 𝜃cos2 𝜃 + tan 𝜃sin2 𝜃
14. sec 2 𝜃 (1 − sin2 𝜃 )
13.)________________
14.)________________
2
Verify the identity.
15. sec 𝜃 cot 𝜃 = csc 𝜃
17.
19.
csc 𝜃
cot 𝜃
= sec 𝜃
sec2 𝜃 − 1
sec2 𝜃
= sin2 𝜃
16. sin 𝜃 ∙
1
csc 𝜃
= sin2 𝜃
18. cos 2 𝜃(1 + tan2 𝜃) = 1
20. tan2 𝜃 ∙ csc 2 𝜃 = sec 2 𝜃
3
Verify the identity.
21.
23.
1
cos 𝜃
− cos 𝜃 = sin 𝜃 tan 𝜃
tan2 𝜃 − 1
tan 𝜃 − 1
22.
= tan 𝜃 + 1
25. (1 − sin 𝜃 )(1 + sin 𝜃 ) =
sin2 𝜃 − cos2 𝜃
sin 𝜃 + cos 𝜃
24. 1 −
1
sec2 𝜃
1
sec2 𝜃
= sin 𝜃 − cos 𝜃
= sin2 𝜃
26. 2cos 2 𝜃 − 1 = 1 − 2sin2 𝜃
4
Find the exact value of the following using sum and difference identities:
27. sin 105°
28. cos 15°
27.)________________
28.)________________
Using reference angles, find the following:
29. cos 120°
sin 120°
30. cos 150°
sin 150°
29.) cos 120°=_______
29.) sin 120°=_______
30.) cos 150°=_______
30.) sin 150°=_______
Now use #29& 30, along with sum and difference identities to find the following:
31. cos 165°
32. sin 195°
[Hint: use (120° + 45°)]
31.)________________
[Hint: use (150° + 45°)]
32.)________________
Answers:
1. cos 𝜃
2. 1
3. cot 2 𝜃
4. tan 𝜃
5. cos 2 𝜃
6. 1
7. 1
8. tan2 𝜃
9. 1
10. cot 2 𝜃
11. 1 − cos 𝜃
12. sec 𝜃
13. tan 𝜃
14. 1
27.
√2 + √6
4
30. cos 150° = −
28.
√3
;
2
√6 + √2
4
sin 150° =
1
29. cos 120° = − 2 ; sin 120° =
1
2
31.
− √2− √6
4
32.
√3
2
√2− √6
4
5