Precalculus by Richard Wright

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Pride brings a person low, but the lowly in spirit gain honor. Proverbs‬ ‭29‬:‭23‬ ‭NIV‬‬

# 5-Review

Take this test as you would take a test in class. When you are finished, check your work against the answers. On this assignment round your answers to three decimal places unless otherwise directed.

1. If tan x = 1 and cos x < 0, find sin x.
2. Simplify (tan2 x + 1)(cot2 x + 1).
3. Simplify sec x − sec x sin2 x.
4. Solve on the interval [0, 2π): sec2 x = 1 + tan x.
5. Verify the identity.

6. $$\cot x = \frac{\csc x \sec x}{1 + \tan^2 x}$$
7. cos(x − π) = −cos x
8. $$\sec\left(x - \frac{π}{2}\right)\cos\left(-x\right) = \cot x$$
9. sin x sin 2x = 2 cos x − 2 cos3 x
10. Use a power reducing formula to rewrite the following in terms of the first power of cosine: sin4 x.
11. If $$\sin x = \frac{\sqrt{3}}{2}$$ and $$0 < x ≤ \frac{π}{2}$$, find $$\tan \frac{x}{2}$$.
12. If $$\sin α = \frac{40}{41}$$ and $$\frac{π}{2} < α < π$$, find tan 2α.
13. Write cos 3x − cos 2x as a product.
14. Write cos 3x sin 2x as a sum or difference.
15. Solve on the interval [0, 2π).

16. cos 3x + cos x = 0
17. sin 2x sec x = 2 sin 2x
18. $$2 \cos x + \sqrt{3} = 0$$
19. $$3 \tan 2x = \sqrt{3}$$
20. 2 cos2 x + 3 cos x + 1 = 0
21. Use a graphing utility to approximate the solutions of tan x + cos x = 0 on the interval [0, 2π). Round to 4 decimal places.
22. Find the exact value of tan 345° given that 345 = 135 + 210.
23. A baseball leaves the hand of the person at first base at an angle of α with the horizontal and at an initial velocity of v0 = 30 meters per second. The ball is caught by another person 20 meters away. Find α if the range, r, of a projectile is $$r = \frac{1}{32} v_0^2 \sin 2α$$. Use degrees.

1. $$-\frac{\sqrt{2}}{2}$$
2. sec2 x csc2 x
3. cos x
4. $$0, \frac{π}{4}, π, \frac{5π}{4}$$
5. $$\cot x = \frac{\csc x}{\sec x}$$ $$= \frac{\csc x \sec x}{\sec^2 x}$$ $$= \frac{\csc x \sec x}{1 + \tan^2 x}$$
6. $$\cos\left(x - π\right)$$ $$= \cos x \cos π + \sin x \sin π$$ $$= -\cos x$$
7. $$\sec\left(x - \frac{π}{2}\right)\cos\left(-x\right)$$ $$= \csc x \left(\cos x\right)$$ $$= \frac{\cos x}{\sin x}$$ $$= \cot x$$
8. $$\sin x \sin 2x$$ $$= 2 \sin x \cos x \sin x$$ $$= 2 \sin^2 x \cos x$$ $$= 2\left(1 - \cos^2 x\right) \cos x$$ $$= 2 \cos x - 2 \cos^3 x$$
9. $$\frac{3 - 4 \cos 2x + \cos 4x}{8}$$
10. $$\frac{\sqrt{3}}{3}$$
11. $$\frac{720}{1519}$$
12. $$-2 \sin \frac{5x}{2} \sin \frac{x}{2}$$
13. $$\frac{1}{2} \left(\sin 5x - \sin x\right)$$
14. $$\frac{π}{4}$$, $$\frac{π}{2}$$, $$\frac{3π}{4}$$, $$\frac{5π}{4}$$, $$\frac{3π}{2}$$, $$\frac{7π}{4}$$
15. 0, $$\frac{π}{3}$$, $$\frac{5π}{3}$$, π
16. $$\frac{5π}{6}$$, $$\frac{7π}{6}$$
17. $$\frac{π}{12}$$, $$\frac{7π}{12}$$, $$\frac{13π}{12}$$, $$\frac{19π}{12}$$
18. $$\frac{2π}{3}$$, π, $$\frac{4π}{3}$$
19. 3.8078, 5.6169
20. $$-2 + \sqrt{3}$$
21. 22.7°, 67.3°