Unit Circle Test – Precalculus Math (Printable PDF)

Question 1 Convert Angle Measures (2 points)

(a) Convert 315° to radians. (1 point) (b) Convert 7π/18 radians to degrees. (1 point)

Question 2 Sprinkler Arc (2 points)

A lawn sprinkler throws water 18 feet and sweeps through an angle of 5π/6 radians.

1. Find the length of the arc the water reaches. (1 point)
2. Find the angle in degrees (nearest tenth) that gives a 30-foot arc at the same radius. (1 point)

Question 3 Point on the Unit Circle (2 points)

The terminal side of θ = 5π/4 meets the unit circle as shown.

1. Give the exact coordinates of the point. (1 point)
2. Find tan θ. (1 point)

Question 4 Exact Trigonometric Values (3 points)

Find the exact value. (a) cos(3π/4) (1 point) (b) sin(−7π/6) (1 point) (c) tan(5π/3) (1 point)

Question 5 Find Cosine and Tangent (2 points)

Angle θ lies in Quadrant III and sin θ = −5/13. Find cos θ and tan θ.

Question 6 Reference Angle and Sine (2 points)

(a) Find the reference angle of 17π/6. (1 point) (b) Find sin(17π/6). (1 point)

Question 7 Tire Rotation (3 points)

A tire of radius 15 inches turns 420 revolutions per minute.

1. Find the angular speed in radians per minute. (1 point)
2. Find the linear speed of the tread in inches per minute. (1 point)
3. Convert that speed to miles per hour (63,360 inches per mile), to the nearest tenth. (1 point)

Question 8 Symmetry of Marked Points (2 points)

Point A is at angle 0.8 radian, with cos 0.8 ≈ 0.697 and sin 0.8 ≈ 0.717. Point B is at −0.8 radian, and point C is at π − 0.8 radian.

1. Find the coordinates of B. (1 point)
2. Find the coordinates of C. (1 point)

Question 9 Particle Motion (2 points)

A point starts at (1, 0) and moves counterclockwise on the unit circle at π/3 radians per second. (a) Find its exact position after 7 seconds. (1 point) (b) When does it first reach (−1, 0)? (1 point)