Precalculus Radians and the Unit Circle Test with Point Values (PDF)

Precalculus math test: Radians and the Unit Circle – Maths PDF

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This angles unit circle test for Precalculus covers radians and the unit circle (Common Core F-TF.A.1, F-TF.A.2, F-TF.A.3, F-TF.A.4), with a free PDF to print.

This 20-point test covers radian measure, arc length, the unit circle, exact values and a Ferris wheel model (F-TF.A.1 to F-TF.A.4). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: Radians and the Unit Circle
Standards: F-TF.A.1, F-TF.A.2, F-TF.A.3, F-TF.A.4
Time: 45 minutes
Total: 20 points

Angles unit circle test: questions

Question 1 – Convert angles [2 pts]

Convert \(135^{\circ}\) to radians and \(\frac{7\pi}{6}\) to degrees.

Question 2 – Coterminal angles [2 pts]

Find one positive and one negative angle coterminal with \(\frac{5\pi}{6}\).

Question 3 – Arc and sector [2 pts]

A circle has radius \(12\) ft. For a central angle of \(\frac{\pi}{4}\), find the arc length and the sector area, to the nearest hundredth.

Question 4 – Exact values [2 pts]

The unit circle with special angles.

Use the unit circle. Find the exact value of \(\sin\frac{5\pi}{6}\) and \(\tan\frac{7\pi}{4}\).

Question 5 – Unit circle point [3 pts]

Point P on the unit circle at (-1/2, √3/2).

The point \(P\left(-\frac12,\frac{\sqrt3}{2}\right)\) lies on the unit circle. (a) Find the angle \(\theta\) in \([0,2\pi)\) in radians. (b) Find \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\). (c) Find the reference angle.

Question 6 – Find the other values [2 pts]

If \(\sin\theta=-\frac{5}{13}\) and \(\theta\) is in quadrant IV, find \(\cos\theta\) and \(\tan\theta\).

Question 7 – Odd and even [2 pts]

Evaluate \(\sin\left(-\frac{\pi}{3}\right)\) and \(\cos\frac{11\pi}{6}\).

Question 8 – Ferris wheel [3 pts]

Ferris wheel of radius 30 feet.

A Ferris wheel of radius \(30\) ft makes one full turn every \(2\) minutes. (a) Find its angular speed in radians per minute. (b) Find the distance a rider travels in \(45\) seconds, to the nearest hundredth. (c) Through how many degrees does it turn in \(45\) seconds?

Question 9 – Solve for an angle [2 pts]

Find all \(\theta\) in \([0,2\pi)\) with \(\cos\theta=-\frac{\sqrt2}{2}\).

Key facts behind this angles unit circle test

Key facts

pi radians = 180 degrees; convert with the factor pi/180 or 180/pi.
Arc length s = r theta and sector area A = (1/2) r^2 theta, with theta in radians.
The point on the unit circle at angle theta is (cos theta, sin theta).
Reference angle: the acute angle to the x-axis; the quadrant sets the sign.
Coterminal angles differ by 2 pi and have the same trig values.

How to use this angles unit circle test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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