Precalculus Practice Test: Trig Identities
- Trig Identities
- 10 questions
- 15 minutes
These 10 questions cover the Pythagorean, reciprocal, sum and double-angle identities, plus simplifying expressions and solving basic trig equations. When simplifying, it often helps to rewrite everything in terms of sine and cosine first.
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Question 1
Which expression is equivalent to sin2 x1 − cos x (where cos x ≠ 1)?
Correct answer: B
Replace sin2 x with 1 − cos2 x = (1 − cos x)(1 + cos x), then cancel 1 − cos x. The result 1 − cos x cancels the wrong factor.
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Question 2
Which expression is equivalent to sec x − tan x sin x?
Correct answer: C
Rewrite as 1/cos x − sin2 x/cos x = (1 − sin2 x)/cos x = cos2 x/cos x = cos x. Stopping at a numerator of 1 gives sec x, which forgets the subtraction.
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Question 3
What is the exact value of cos 75°?
Correct answer: A
Use cos(45° + 30°) = cos 45° cos 30° − sin 45° sin 30° = √6/4 − √2/4. The sum (√6 + √2)/4 is cos 15°, which uses a plus sign in the formula.
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Question 4
Angle θ is in Quadrant II and sin θ = 35. What is sin 2θ?
Correct answer: D
In Quadrant II cos θ = −4/5, so sin 2θ = 2 sin θ cos θ = 2(3/5)(−4/5) = −24/25. The value 24/25 uses a positive cosine, and 7/25 is cos 2θ.
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Question 5
Which expression is equivalent to cos2 x − sin2 x?
Correct answer: A
This is one form of the double-angle identity for cosine. The value 1 is cos2 x + sin2 x, with a plus sign instead of a minus.
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Question 6
What are the solutions of 2 sin x − 1 = 0 on the interval [0, 2π)?
Correct answer: C
The equation gives sin x = 1/2, which happens in Quadrants I and II at π/6 and 5π/6. In Quadrants III and IV the sine is negative, so 7π/6 and 11π/6 give −1/2.
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Question 7
Which expression is equivalent to tan x + cot x?
Correct answer: B
Write sin x/cos x + cos x/sin x = (sin2 x + cos2 x)/(sin x cos x) = 1/(sin x cos x) = sec x csc x. The product sin x cos x is the reciprocal of the correct answer.
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Question 8
Which expression is equal to 1 + tan2 x for every x where both sides are defined?
Correct answer: D
Divide sin2 x + cos2 x = 1 by cos2 x to get tan2 x + 1 = sec2 x. The identity with csc2 x is 1 + cot2 x.
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Question 9
What are all the solutions of cos2 x = cos x on the interval [0, 2π)?
Correct answer: B
Move everything to one side: cos x(cos x − 1) = 0, so cos x = 0 (π/2, 3π/2) or cos x = 1 (0). Dividing both sides by cos x loses π/2 and 3π/2.
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Question 10
Which expression is equivalent to sin(x + π)?
Correct answer: A
By the sum formula, sin x cos π + cos x sin π = sin x(−1) + cos x(0) = −sin x. Adding π moves a point to the opposite side of the unit circle, which changes the sign of sine.
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