Precalculus Practice Test: Unit Circle and Trig Ratios
- Unit Circle and Trig Ratios
- 10 questions
- 15 minutes
These 10 questions cover radian measure, exact values on the unit circle, reference and coterminal angles, and trig ratios from a point. To get the sign right, remember which ratios are positive in each quadrant: all in I, sine in II, tangent in III, cosine in IV.
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Question 1
What is 150° in radians?
Correct answer: A
Multiply by π/180: 150 · π/180 = 5π/6. The angle 2π/3 is 120°, and 5π/3 is 300°.
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Question 2
What is the exact value of sin(7π6)?
Correct answer: C
The angle 7π/6 is in Quadrant III with reference angle π/6, and sine is negative there, so the value is −1/2. The value −√3/2 is cos(7π/6).
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Question 3
What is the exact value of cos(5π3)?
Correct answer: B
The angle 5π/3 is in Quadrant IV with reference angle π/3, where cosine is positive, so cos(5π/3) = cos(π/3) = 1/2. A negative answer uses the sign of sine in that quadrant.
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Question 4
What is the exact value of tan(3π4)?
Correct answer: D
The angle 3π/4 is in Quadrant II, where sine is positive and cosine is negative, and its reference angle is π/4. So the tangent is −(√2/2)/(√2/2) = −1.
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Question 5
Angle θ is in Quadrant III and sin θ = −35. What is cos θ?
Correct answer: B
From sin2 θ + cos2 θ = 1, cos2 θ = 1 − 9/25 = 16/25, so cos θ = ±4/5, and cosine is negative in Quadrant III. The value 3/4 is tan θ.
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Question 6
What is the reference angle of 230°?
Correct answer: A
An angle in Quadrant III has reference angle θ − 180°, so 230° − 180° = 50°. The value 40° measures to the y-axis (270°), but reference angles are always measured to the x-axis.
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Question 7
Which angle is coterminal with −100°?
Correct answer: D
Coterminal angles differ by a multiple of 360°: −100° + 360° = 260°. The angle 100° only removes the sign, which reflects the terminal side instead of keeping it.
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Question 8
The point (−5, 12) is on the terminal side of angle θ. What is sec θ?
Correct answer: C
The distance to the origin is r = 25 + 144 = 13, and sec θ = r/x = 13/(−5). The value −5/13 is cos θ, the reciprocal of the secant.
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Question 9
A circle has radius 6 cm. What is the length of the arc cut off by a central angle of 2π3 radians?
Correct answer: A
Arc length is s = rθ = 6 · 2π/3 = 4π cm. The answer 12π cm multiplies the radius by 2π and forgets the denominator 3.
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Question 10
Which of these values is undefined?
Correct answer: B
At π/2 the unit circle point is (0, 1), so tan(π/2) = 1/0, which is undefined. The cotangent is 0/1 = 0 there, so it is defined.
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