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Math 30-1 · Worksheets

Mixed review · All trigonometry

Ten questions of mixed difficulty, covering Sinusoidal models, Trig identities, Trigonometry. Print it, or work through it on screen — the answer key starts on its own page.

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Mixed review · All trigonometry

Math 30-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Which expression is equal to tan x?

    1. a) cos x/sin x
    2. b) sin x/cos x
    3. c) 1/sin x
    4. d) sin x · cos x

  2. Using a sum identity, what is the exact value of cos 105°?

    1. a) (√2 − √6)/4
    2. b) (√6 − √2)/4
    3. c) (√2 + √6)/4
    4. d) −1/2

  3. What is the non-permissible value of sin x ÷ (1 − cos x) for 0 ≤ x < 2π?

    1. a) there is none
    2. b) x = 0
    3. c) x = π/2 and x = 3π/2
    4. d) x = π

  4. Solve sin 2x = sin x for 0 ≤ x < 2π.

    1. a) π/6, 5π/6
    2. b) π/3, 5π/3
    3. c) 0, π
    4. d) 0, π/3, π, 5π/3

  5. Solve sin θ = 1/2 for 0 ≤ θ < 2π.

    1. a) π/6 and 7π/6
    2. b) π/6 and 5π/6
    3. c) π/3 and 2π/3
    4. d) π/6 only

  6. sin²x + cos²x equals:

    1. a) sec²x
    2. b) 0
    3. c) 1
    4. d) tan²x

  7. Simplify (1 − sin²x) ÷ cos x.

    1. a) cos x
    2. b) sec x
    3. c) sin x
    4. d) 1

  8. A sinusoidal graph has a maximum of 9 and a minimum of 1. What is the equation of its midline?

    1. a) y = 5
    2. b) y = 0
    3. c) y = 4
    4. d) y = 8

  9. If cos x = 1/3, what is cos 2x?

    1. a) 7/9
    2. b) −7/9
    3. c) 2/3
    4. d) 1/9

  10. What is 3π/4 in degrees?

    1. a) 270°
    2. b) 135°
    3. c) 120°
    4. d) 150°

Answer key · Mixed review · All trigonometry

Math 30-1 · maddyhelps.com

  1. b) sin x/cos x — tan x = y/x on the unit circle, and y = sin x while x = cos x. cos x/sin x is cot x.
  2. a) (√2 − √6)/4 — cos(60° + 45°) = cos 60° cos 45° − sin 60° sin 45° = (1/2)(√2/2) − (√3/2)(√2/2) = (√2 − √6)/4. It is negative, as it must be in Quadrant II.
  3. b) x = 0 — The denominator is zero when cos x = 1, which happens at x = 0 in this domain. In general, x ≠ 2πn.
  4. d) 0, π/3, π, 5π/3 — Rewrite: 2 sin x cos x − sin x = 0, so sin x(2cos x − 1) = 0. sin x = 0 gives 0 and π; cos x = 1/2 gives π/3 and 5π/3. Dividing by sin x would lose half the answers.
  5. b) π/6 and 5π/6 — The reference angle is π/6, and sine is positive in Quadrants I and II: π/6 and π − π/6 = 5π/6. Stopping at one answer is the usual slip.
  6. c) 1 — On the unit circle, sin x and cos x are the legs of a right triangle with hypotenuse 1, so the Pythagorean theorem gives sin²x + cos²x = 1.
  7. a) cos x — 1 − sin²x = cos²x from the Pythagorean identity, and cos²x ÷ cos x = cos x.
  8. a) y = 5 — The midline sits halfway between the extremes: (9 + 1) ÷ 2 = 5.
  9. b) −7/9 — Use the form of cos 2x written only in cosine: 2cos²x − 1 = 2(1/9) − 1 = −7/9. 2/3 wrongly doubles the cosine.
  10. b) 135° — Replace π with 180°: 3 × 180° ÷ 4 = 135°.