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Math 30-1 · Worksheets

Trig identities · Set A

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

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Trig identities · Set A

Math 30-1 · maddyhelps.com

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

  1. What is the exact value of 2 sin 15° cos 15°?

    1. a) √2/2
    2. b) 1/4
    3. c) 1/2
    4. d) √3/2

  2. Simplify csc x · tan x.

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

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

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

  4. Simplify sec x · cos x.

    1. a) tan x
    2. b) 1
    3. c) cos²x
    4. d) sec²x

  5. When proving a trigonometric identity, you should:

    1. a) substitute one value of x and check both sides
    2. b) square both sides first
    3. c) move terms from one side to the other, as when solving an equation
    4. d) work on each side separately until both sides match

  6. Simplify (1 − sin²x) ÷ cos x.

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

  7. Which expression is equal to sec x?

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

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

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

  9. Which of these is a true identity?

    1. a) 1 − tan²x = sec²x
    2. b) 1 + tan²x = csc²x
    3. c) tan²x − 1 = sec²x
    4. d) 1 + tan²x = sec²x

  10. sin A = 3/5 and cos B = 5/13, with A and B both in Quadrant I. What is sin(A + B)?

    1. a) 56/65
    2. b) 63/65
    3. c) −16/65
    4. d) 33/65

Answer key · Trig identities · Set A

Math 30-1 · maddyhelps.com

  1. c) 1/2 — Recognise the double-angle pattern: 2 sin 15° cos 15° = sin 30° = 1/2. Spotting the identity saves any calculation.
  2. d) sec x — Write everything in sine and cosine: (1/sin x) · (sin x/cos x) = 1/cos x = sec x.
  3. 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.
  4. b) 1 — sec x = 1/cos x, so (1/cos x) · cos x = 1. A function times its reciprocal is always 1.
  5. d) work on each side separately until both sides match — An identity is what you are trying to prove, so you cannot treat it as a true equation yet. Simplify each side on its own until they are identical.
  6. d) cos x — 1 − sin²x = cos²x from the Pythagorean identity, and cos²x ÷ cos x = cos x.
  7. b) 1/cos x — Secant is the reciprocal of cosine. A quick way to remember: each reciprocal pairs with the function whose name starts with a different letter — sec with cos, csc with sin.
  8. a) −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.
  9. d) 1 + tan²x = sec²x — Divide sin²x + cos²x = 1 by cos²x and you get tan²x + 1 = sec²x. Dividing by sin²x instead gives 1 + cot²x = csc²x.
  10. b) 63/65 — From 3-4-5 and 5-12-13 triangles, cos A = 4/5 and sin B = 12/13. Then sin(A + B) = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65.