NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-4 Practice
Sum and Difference Identities
Find the exact value of each trigonometric expression.
5𝜋
1. cos 12
2. sin (–165°)
4. csc 915°
5. tan (−
7. cos (–15°)
8. tan 15°
9. cos (−
11. sin 20° cos 10° + cos 20° sin 10°
12.
10. cos
11𝜋
12
3. tan 345°
7𝜋
)
12
6. sec
𝜋
12
7𝜋
)
12
π
5𝜋
+ tan
9
36
π
5𝜋
1 − tan tan
9
36
tan
Simplify each expression.
13. cos
3𝜋
2
cos π – sin
3𝜋
2
sin π
15. cos 70° cos 20° – sin 70° sin 20°
tan 30° − tan 𝑥
14. 1 + tan 30° tan 𝑥
𝜋
𝜋
16. sin 12 cos y – sin y cos 12
Write each trigonometric expression as an algebraic expression.
1
2
17. sin (arccos x + arcsin x)
18. cos (arccos
− arcsin 𝑥)
19. cos (arcsin x + arccos x)
20. sin (arctan 3 – arcsin x)
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Verify each cofunction or reduction identity.
21. sin (360° + θ) = sin θ
22. cos (180° – θ) = –cos θ
Find the solutions to each expression on the interval [0, 2π).
5π
5π
23. cos ( 4 + 𝑥) + sin ( 4 − 𝑥) = 0
𝜋
𝜋
25. cos ( 4 − 𝑥) – sin ( 4 − 𝑥) = –1
3𝜋
3𝜋
27. cos ( 2 + 𝑥) + sin ( 2 − 𝑥) = 0
2π
2π
24. sin ( 3 − 𝑥) + sin ( 3 + 𝑥) = 0
26. sin (π + x) + sin (π + x) = 1
28. tan (π – x) + tan (π – x) = –2
29. Sound waves can be modeled by the equations of the form 𝑦1 = 20 sin (3x + θ). A wave traveling in the opposite
direction can be modeled by 𝑦2 = 20 sin (3x – θ). Show that 𝑦1 + 𝑦2 = 40 sin 3x cos θ.
30. ENGINEERING Two highways branch off each other at an angle of 75°. An engineer uses tan 75° to determine the
height of an exit ramp at a particular point. Find the exact value of tan 75°.
31. ELECTRICITY The current I in amperes in an alternating current at time t in seconds can be found with the formula
7𝜋
I = 30 sin (50𝜋𝑡 − 3 ). Rewrite the formula in terms of one or more functions of 50πt.
32. TEMPERATURE A city’s average daily high temperature can be modeled by y = 14.33
sin (0.56x – 2.44) + 60.79, where x = 1 corresponds to January. Rewrite the formula using the Sine Difference Identity.