Precalculus Radians and the Unit Circle Test Solutions and Scoring (PDF)
This angles unit circle test 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.
Precalculus · Common Core State Standards: F-TF.A.1, F-TF.A.2, F-TF.A.3, F-TF.A.4
Angles unit circle test answers: worked solutions
Question 1 – Convert angles [2 pts]
\(135\cdot\frac{\pi}{180}=\frac{3\pi}{4}\). \(\frac{7\pi}{6}\cdot\frac{180}{\pi}=210^{\circ}\).
\(\frac{3\pi}{4}\) and \(210^{\circ}\)
Scoring: 1 point each.
Question 2 – Coterminal angles [2 pts]
\(\frac{5\pi}{6}+2\pi=\frac{17\pi}{6}\). \(\frac{5\pi}{6}-2\pi=-\frac{7\pi}{6}\).
\(\frac{17\pi}{6}\) and \(-\frac{7\pi}{6}\)
Scoring: 1 point each.
Question 3 – Arc and sector [2 pts]
\(s=12\cdot\frac{\pi}{4}=3\pi\approx 9.42\). \(A=\frac12(144)\frac{\pi}{4}=18\pi\approx 56.55\).
\(9.42\text{ ft}\); \(56.55\text{ ft}^{2}\)
Scoring: 1 point each.
Question 4 – Exact values [2 pts]
\(\frac{5\pi}{6}\) has reference angle \(\frac{\pi}{6}\) in quadrant II: \(\frac12\). \(\frac{7\pi}{4}\) has reference \(\frac{\pi}{4}\) in quadrant IV, where tangent is negative: \(-1\).
\(\frac12\) and \(-1\)
Scoring: 1 point each.
Question 5 – Unit circle point [3 pts]
(a) The point is in quadrant II with \(\cos\theta=-\frac12\), so \(\theta=\frac{2\pi}{3}\). (b) \(\sin=\frac{\sqrt3}{2}\), \(\cos=-\frac12\), \(\tan=-\sqrt3\). (c) \(\pi-\frac{2\pi}{3}=\frac{\pi}{3}\).
\(\frac{2\pi}{3}\); \(\frac{\sqrt3}{2},\ -\frac12,\ -\sqrt3\); \(\frac{\pi}{3}\)
Scoring: 1 point for each part.
Question 6 – Find the other values [2 pts]
\(\cos^{2}\theta=1-\frac{25}{169}=\frac{144}{169}\), positive in quadrant IV: \(\frac{12}{13}\). \(\tan\theta=-\frac{5}{12}\).
\(\cos\theta=\frac{12}{13},\ \tan\theta=-\frac{5}{12}\)
Scoring: 1 point each.
Question 7 – Odd and even [2 pts]
\(\sin\left(-\frac{\pi}{3}\right)=-\frac{\sqrt3}{2}\). \(\frac{11\pi}{6}\) has reference \(\frac{\pi}{6}\) in quadrant IV: \(\cos=\frac{\sqrt3}{2}\).
\(-\frac{\sqrt3}{2}\) and \(\frac{\sqrt3}{2}\)
Scoring: 1 point each.
Question 8 – Ferris wheel [3 pts]
(a) \(\frac{2\pi}{2}=\pi\) rad/min. (b) \(45\text{ s}=0.75\text{ min}\), so \(\theta=0.75\pi\) and \(s=30\cdot0.75\pi=22.5\pi\approx 70.69\). (c) \(0.75\pi\) rad \(=135^{\circ}\).
\(\pi\) rad/min; \(70.69\text{ ft}\); \(135^{\circ}\)
Scoring: 1 point for each part.
Question 9 – Solve for an angle [2 pts]
Cosine is negative in quadrants II and III with reference angle \(\frac{\pi}{4}\): \(\pi-\frac{\pi}{4}\) and \(\pi+\frac{\pi}{4}\).
\(\theta=\frac{3\pi}{4},\ \frac{5\pi}{4}\)
Scoring: 1 point for each angle.
Total score
20 points
Key facts behind this angles unit circle test answers
- 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 answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
Download the angles unit circle test answers as a PDF with the button below and keep it with your notes.
All Precalculus chapters: Precalculus math.
