Precalculus Radians and the Unit Circle Worksheet Solutions (PDF)

Precalculus math worksheet solutions: Radians and the Unit Circle – Maths PDF

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This angles unit circle answers 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.

Use this key to check each angle or trig value. Solutions show the conversion factor, the reference angle and the quadrant sign, with final answers in bold.

Precalculus · Common Core State Standards: F-TF.A.1, F-TF.A.2, F-TF.A.3, F-TF.A.4

Angles unit circle answers: exercises 1 to 10

Exercise 1 – Degrees to radians

\(45\cdot\frac{\pi}{180}=\frac{\pi}{4}\). \(150\cdot\frac{\pi}{180}=\frac{5\pi}{6}\).

\(\frac{\pi}{4}\) and \(\frac{5\pi}{6}\)

Exercise 2 – Radians to degrees

Multiply by \(\frac{180}{\pi}\): \(30^{\circ}\) and \(225^{\circ}\).

\(30^{\circ}\) and \(225^{\circ}\)

Exercise 3 – Calculator conversion

\(3\cdot\frac{180}{\pi}\approx 171.89^{\circ}\). \(100\cdot\frac{\pi}{180}\approx 1.75\).

\(171.89^{\circ}\) and \(1.75\) rad

Exercise 4 – Coterminal angles

\(50+360=410^{\circ}\); \(50-360=-310^{\circ}\). For \(\frac{7\pi}{4}\): \(\frac{7\pi}{4}-2\pi=-\frac{\pi}{4}\); \(\frac{7\pi}{4}+2\pi=\frac{15\pi}{4}\).

\(410^{\circ},\ -310^{\circ}\); \(\frac{15\pi}{4},\ -\frac{\pi}{4}\)

Exercise 5 – Reference angle

\(180^{\circ}-150^{\circ}=30^{\circ}\). \(\frac{5\pi}{4}-\pi=\frac{\pi}{4}\).

\(30^{\circ}\) and \(\frac{\pi}{4}\)

Exercise 6 – Quadrants

200 degrees is in quadrant III; the other two are in quadrant IV.

\(200^{\circ}\) is between \(180^{\circ}\) and \(270^{\circ}\). \(-45^{\circ}\) turns clockwise into the lower right. \(\frac{11\pi}{6}=330^{\circ}\).

III, IV, IV

Exercise 7 – Arc length

\(s=r\theta=6\cdot\frac{\pi}{3}=2\pi\approx6.28\).

\(2\pi\approx6.28\text{ in}\)

Exercise 8 – Sector area

\(A=\frac12(10)^{2}\cdot\frac{2\pi}{5}=20\pi\approx 62.83\).

\(20\pi\approx 62.83\text{ ft}^{2}\)

Exercise 9 – Find the angle

\(\theta=\frac{s}{r}=\frac{15}{6}=2.5\) rad. In degrees: \(2.5\cdot\frac{180}{\pi}\approx 143.2^{\circ}\).

\(2.5\) rad \(\approx 143.2^{\circ}\)

Exercise 10 – Exact values, first quadrant

\(\sin30^{\circ}=\frac12\). \(\cos45^{\circ}=\frac{\sqrt2}{2}\). \(\tan60^{\circ}=\frac{\sqrt3/2}{1/2}=\sqrt3\).

\(\frac12,\ \frac{\sqrt2}{2},\ \sqrt3\)

Exercises 11 and up

Exercise 11 – Exact values, other quadrants

\(\frac{2\pi}{3}\): reference \(\frac{\pi}{3}\), quadrant II, cosine negative: \(-\frac12\). \(\frac{5\pi}{4}\): reference \(\frac{\pi}{4}\), quadrant III: \(-\frac{\sqrt2}{2}\). \(\frac{3\pi}{4}\): quadrant II: \(\tan=\frac{\sqrt2/2}{-\sqrt2/2}=-1\).

\(-\frac12,\ -\frac{\sqrt2}{2},\ -1\)

Exercise 12 – Point on the unit circle

The point is (-√3/2, 1/2).

The reference angle is \(\frac{\pi}{6}\) in quadrant II, so \(\cos\theta=-\frac{\sqrt3}{2}\) and \(\sin\theta=\frac12\). \(\tan\theta=\frac{1/2}{-\sqrt3/2}=-\frac{1}{\sqrt3}=-\frac{\sqrt3}{3}\).

\(\left(-\frac{\sqrt3}{2},\frac12\right)\); \(\tan\theta=-\frac{\sqrt3}{3}\)

Exercise 13 – Find the other values

\(\cos^{2}\theta=1-\frac{9}{25}=\frac{16}{25}\). In quadrant II cosine is negative: \(-\frac45\). \(\tan\theta=\frac{3/5}{-4/5}\).

\(\cos\theta=-\frac45,\ \tan\theta=-\frac34\)

Exercise 14 – Another quadrant

\(\sin^{2}\theta=1-\frac{25}{169}=\frac{144}{169}\). Quadrant III: sine is negative, \(-\frac{12}{13}\). \(\tan\theta=\frac{-12/13}{-5/13}\).

\(\sin\theta=-\frac{12}{13},\ \tan\theta=\frac{12}{5}\)

Exercise 15 – Negative angles

Sine is odd: \(-\sin\frac{\pi}{6}=-\frac12\). Cosine is even: \(\cos\frac{\pi}{3}=\frac12\).

\(-\frac12\) and \(\frac12\)

Exercise 16 – Large and negative angles

\(\frac{7\pi}{3}-2\pi=\frac{\pi}{3}\), so \(\sin=\frac{\sqrt3}{2}\). \(\cos\left(-\frac{5\pi}{6}\right)=\cos\frac{5\pi}{6}=-\frac{\sqrt3}{2}\).

\(\frac{\sqrt3}{2}\) and \(-\frac{\sqrt3}{2}\)

Exercise 17 – Ferris wheel

Speed pi/10 radians per second.

\(3\cdot2\pi=6\pi\) rad per minute, so \(\frac{6\pi}{60}=\frac{\pi}{10}\) rad/s. In 10 s: \(\theta=\pi\), so \(s=25\pi\approx 78.5\).

\(\frac{\pi}{10}\) rad/s; \(25\pi\approx 78.5\text{ ft}\)

Exercise 18 – Pizza slice

One slice.

\(\theta=\frac{2\pi}{8}=\frac{\pi}{4}\). Crust: \(s=7\cdot\frac{\pi}{4}\approx 5.5\). Area: \(\frac12(49)\frac{\pi}{4}=\frac{49\pi}{8}\approx 19.24\).

\(5.5\text{ in}\); \(19.24\text{ in}^{2}\)

Exercise 19 – Windshield wiper

\(120^{\circ}=\frac{2\pi}{3}\). \(A=\frac12(18)^{2}\cdot\frac{2\pi}{3}=108\pi\approx 339.3\).

\(108\pi\approx 339.3\text{ in}^{2}\)

Exercise 20 – Distance on Earth

\(5^{\circ}=\frac{5\pi}{180}=\frac{\pi}{36}\) rad. \(s=3959\cdot\frac{\pi}{36}\approx 345\).

About \(345\) miles

Key facts behind this angles unit circle answers

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 answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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