Precalculus Unit Circle Quiz: 10 Questions

Contents

Key takeaway: This precalculus unit circle quiz asks ten multiple-choice questions about radians, special-angle points, reference angles, arc length, angular speed and symmetry. Only one option fits each question, and each note flags the typical error. Standard F-TF.A.1 guides the items, and three questions include a figure. Finish in about 15 minutes, then treat the score as a checkpoint ahead of the full exam.

Ten multiple-choice questions on radian conversion, special angles, reference angles, arc length, trigonometric equations and even, odd and periodic behavior. Choose one answer for each question, then read the explanations to see the typical slips.

PrecalculusCommon Core: F-TF.A.1 · F-TF.A.2 · F-TF.A.3 · F-TF.A.4

🎯 Angles, Radians and the Unit Circle – Precalculus math quiz

Score: 0 / 10

Answered: 0 / 10

Question 1

What is \(240^{\circ}\) in radians?

A \(\frac{4\pi}{3}\)

B \(\frac{2\pi}{3}\)

C \(\frac{5\pi}{3}\)

D \(\frac{3\pi}{4}\)

Question 2

Angle 5pi/4 on the unit circle.

Which are the coordinates of point \(P\)?

A \(\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\)

B \(\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)\)

C \(\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\)

D \(\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)\)

Question 3

What is the reference angle of \(\frac{5\pi}{3}\)?

A \(\frac{\pi}{3}\)

B \(\frac{2\pi}{3}\)

C \(\frac{\pi}{6}\)

D \(\frac{5\pi}{6}\)

Question 4

What is the exact value of \(\cos\left(-\frac{\pi}{3}\right)\)?

A \(-\frac{1}{2}\)

B \(\frac{1}{2}\)

C \(\frac{\sqrt{3}}{2}\)

D \(-\frac{\sqrt{3}}{2}\)

Question 5

A sector with radius 8 ft and a 135 degree central angle.

What is the arc length of the sector?

A \(\frac{3\pi}{4}\text{ ft}\)

B \(6\pi\text{ ft}\)

C \(\frac{3\pi}{32}\text{ ft}\)

D \(1080\text{ ft}\)

Question 6

A wheel turns 5 revolutions per second. What is its angular speed?

A \(5\pi\text{ rad/s}\)

B \(5\text{ rad/s}\)

C \(10\pi\text{ rad/s}\)

D \(20\pi\text{ rad/s}\)

Question 7

Which pair lists every solution of \(\sin\theta=-\frac{1}{2}\) in \([0,2\pi)\)?

A \(\frac{\pi}{6}\text{ and }\frac{5\pi}{6}\)

B \(\frac{5\pi}{6}\text{ and }\frac{7\pi}{6}\)

C \(\frac{7\pi}{6}\text{ and }\frac{11\pi}{6}\)

D \(\frac{\pi}{6}\text{ and }\frac{11\pi}{6}\)

Question 8

The sine graph is unchanged by a half-turn about the origin.

Which equation states the symmetry of this graph about the origin?

A \(\sin(-x)=\sin x\)

B \(\cos(-x)=-\cos x\)

C \(\sin(x+\pi)=\sin x\)

D \(\sin(-x)=-\sin x\)

Question 9

Which function is even?

A \(f(x)=\sin x\)

B \(f(x)=\tan x\)

C \(f(x)=x\sin x\)

D \(f(x)=x\cos x\)

Question 10

The point \(\left(\frac{3}{5},y\right)\) lies on the unit circle in Quadrant IV. What is \(y\)?

A \(\frac{4}{5}\)

B \(-\frac{16}{25}\)

C \(-\frac{2}{5}\)

D \(-\frac{4}{5}\)

🎉 Quiz complete!

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Remember:

180 degrees = pi radians; multiply degrees by pi/180 to get radians.
The point at angle theta on the unit circle is (cos theta, sin theta), and tan theta = sin theta / cos theta.
Special values: sin and cos of pi/6, pi/4, pi/3 are 1/2, sqrt2/2, sqrt3/2 in some order.
Arc length is s = r*theta and linear speed is v = r*omega, with theta in radians.
Sine and cosine have period 2pi, tangent has period pi; cosine is even, sine and tangent are odd.

What do the first four questions test?

Question 1 converts \(240^{\circ}\) to \(\frac{4\pi}{3}\). In question 2, the angle \(\frac{5\pi}{4}\) gives a point with both coordinates \(-\frac{\sqrt{2}}{2}\). Next, the reference angle of \(\frac{5\pi}{3}\) is \(\frac{\pi}{3}\) in question 3. Finally, question 4 asks for \(\cos(-\frac{\pi}{3})=\frac{1}{2}\).

How do you solve the arc and speed questions?

Question 5 shows a sector of radius 8 ft and \(135^{\circ}\), which gives \(6\pi\) ft. Next, question 6 multiplies 5 revolutions per second by \(2\pi\) to get \(10\pi\) rad/s. A common wrong choice forgets the \(2\pi\).

How can you retake the precalculus unit circle quiz?

Cover the choices and sketch the circle first. After that, compare with the options. For question 7, \(\sin\theta=-\frac{1}{2}\) has solutions \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\), because the sine value dips below zero only in the lower half of the circle.

What do questions 8 to 10 check?

Question 8 names the odd symmetry \(\sin(-x)=-\sin x\). Then \(f(x)=x\sin x\) is the even choice in question 9, since both factors flip sign together. Finally, question 10 gives \(y=-\frac{4}{5}\).

What is a smart follow-up to this circle quiz?

Retry the questions you missed after a night of sleep. Then draw the unit circle once more and label each point. If you score 9 or more, go straight to the timed test.