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.

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

    14. Solve on the interval [0, 2π).

  14. cos 3x + cos x = 0
  15. sin 2x sec x = 2 sin 2x
  16. \(2 \cos x + \sqrt{3} = 0\)
  17. \(3 \tan 2x = \sqrt{3}\)
  18. 2 cos2 x + 3 cos x + 1 = 0
  19. Use a graphing utility to approximate the solutions of tan x + cos x = 0 on the interval [0, 2π). Round to 4 decimal places.
  20. Find the exact value of tan 345° given that 345 = 135 + 210.
  21. 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.

Answers

  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°