Angles, Radians and the Unit Circle: Precalculus Math Test Answer Key – PDF Download

Precalculus Unit Circle Test Answer Key: questions 1–5

Question 1: Converting angle measures (2 points)

1. \(225\cdot\frac{\pi}{180}=\)\(\frac{5\pi}{4}\)
2. \(\frac{7\pi}{6}\cdot\frac{180^{\circ}}{\pi}=\)\(210^{\circ}\)

Scoring: 1 point for each correct conversion.

Question 2: Coterminal and reference angles (2 points)

1. \(\frac{23\pi}{6}-2\pi=\)\(\frac{11\pi}{6}\)
2. The angle \(\frac{11\pi}{6}\) is in Quadrant IV, and \(2\pi-\frac{11\pi}{6}=\)\(\frac{\pi}{6}\)

Scoring: 1 point for each part; award 0.5 for a correct method with an arithmetic slip.

Question 3: Point on the unit circle (3 points)

1. \(P=(\cos\frac{7\pi}{6},\sin\frac{7\pi}{6})=\)\((-\frac{\sqrt{3}}{2},-\frac{1}{2})\)
2. \(\tan\frac{7\pi}{6}=\frac{-1/2}{-\sqrt{3}/2}=\frac{1}{\sqrt{3}}=\)\(\frac{\sqrt{3}}{3}\)
3. \(\sin(-\frac{7\pi}{6})=-\sin\frac{7\pi}{6}=-(-\frac{1}{2})=\)\(\frac{1}{2}\)

Scoring: 1 point per part: for (a) 0.5 for each correct coordinate; for (b) 0.5 for the quotient setup and 0.5 for rationalizing; for (c) 0.5 for using the odd property and 0.5 for the value.

Question 4: Arc length of a sector (2 points)

1. \(150^{\circ}=\)\(\frac{5\pi}{6}\)
2. \(s=15\cdot\frac{5\pi}{6}=12.5\pi\approx\) 39.27 in

Scoring: 1 point for each part.

Question 5: Angular and linear speed (3 points)

1. \(\omega=\frac{2\pi}{8}=\)\(\frac{\pi}{4}\text{ rad/min}\)
2. \(v=r\omega=50\cdot\frac{\pi}{4}=12.5\pi\approx\) 39.27 ft/min
3. \(12.5\pi\cdot 20=250\pi\approx\) 785.40 ft

Scoring: 1 point for each part.

Precalculus Unit Circle Test Answer Key: questions 6–9

Question 6: Solving trigonometric equations (2 points)

1. The reference angle is \(\frac{\pi}{6}\). The value is negative, so the terminal side is in Quadrant II or III: \(\theta=\pi-\frac{\pi}{6}=\)\(\frac{5\pi}{6}\) and \(\theta=\pi+\frac{\pi}{6}=\)\(\frac{7\pi}{6}\).
2. The reference angle is \(\frac{\pi}{4}\). The value is positive, so the terminal side is in Quadrant I or III: \(\theta=\)\(\frac{\pi}{4}\) and \(\theta=\pi+\frac{\pi}{4}=\)\(\frac{5\pi}{4}\).

Scoring: 1 point for each part (0.5 for each correct angle).

Question 7: Even and odd (2 points)

1. \(\tan(-\frac{5\pi}{4})=-\tan\frac{5\pi}{4}=-1\), so -1
2. \(f(-x)=(-x)^{2}\sin(-x)=-x^{2}\sin x=-f(x)\), so f is odd

Scoring: 1 point for each part; part (b) needs the computation of f(-x).

Question 8: Periodicity of sine (2 points)

1. The graph repeats every \(2\pi\).
2. \(\sin\frac{13\pi}{6}=\sin(\frac{\pi}{6}+2\pi)=\sin\frac{\pi}{6}=\)\(\frac{1}{2}\). Also \(-\frac{7\pi}{6}+2\pi=\frac{5\pi}{6}\) and \(\sin\frac{5\pi}{6}=\)\(\frac{1}{2}\)

Scoring: 1 point for the period; 1 point for both values (0.5 each).

Question 9: Unit circle coordinates (2 points)

1. \(\sin^{2}\theta=1-\frac{64}{289}=\frac{225}{289}\); sine is positive in Quadrant II, so \(\sin\theta=\frac{15}{17}\)
2. \(\tan\theta=\frac{15/17}{-8/17}=\)\(-\frac{15}{8}\)

Scoring: 1 point for each part; part (b) can be earned with an incorrect part (a) if the quotient is correct.

Scoring summary

How to use this precalculus unit circle test answer key

Use this precalculus unit circle test answer key to score the test fairly and to see which skills need more practice.

Printable PDF · Test Answer Key

Precalculus Angles, Radians and the Unit Circle test answer key

• 📄 US Letter
• 💾 185 KB
• 🖨️ Black and white friendly

Download the PDF

See every chapter on the Precalculus math page.

📘 The Mathovore bookPrecalculus Math Tests with Answer KeysA 45-minute test for every chapter

• 18 chapters in one PDF
• Tests scored out of 20 points
• Answer keys with scoring guides
• Perfect to prepare for exams

See the book →