Radians and the Unit Circle: Algebra 2 Math Test Answer Key – PDF Download

Question 1: Converting angle measures (2 points)

1. \(150\cdot\frac{\pi}{180}=\)\(\frac{5\pi}{6}\).
2. \(\frac{7\pi}{4}\cdot\frac{180^{\circ}}{\pi}=\)\(315^{\circ}\).

Scoring: 1 point for each correct conversion.

Question 2: Arc length and angle (2 points)

1. \(s=9\cdot\frac{2\pi}{3}=\)\(6\pi\) in.
2. \(\theta=\frac{12}{8}=\)\(1.5\) radians.

Scoring: 1 point for each part; accept \(18.85\) inches for part a if the decimal is correct.

Question 3: Point on the unit circle (2 points)

Reference angle \(\frac{\pi}{6}\) in Quadrant III, where both coordinates are negative: \((-\frac{\sqrt{3}}{2},\ -\frac{1}{2})\).

Scoring: 1 point for the reference angle or correct magnitudes; 1 point for both signs and the final point.

Question 4: Exact sine and cosine (2 points)

1. Quadrant IV, reference \(\frac{\pi}{3}\), sine negative: \(-\frac{\sqrt{3}}{2}\).
2. Quadrant II, reference \(\frac{\pi}{4}\), cosine negative: \(-\frac{\sqrt{2}}{2}\).

Scoring: 1 point for each correct value with its sign.

Question 5: Tangent values (2 points)

1. \(\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}\) and \(\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\): \(-1\).
2. \(\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}\) and \(\cos\frac{2\pi}{3}=-\frac{1}{2}\): \(-\sqrt{3}\).

Scoring: 1 point for each value.

Question 6: Solving a sine equation (2 points)

Reference angle \(\frac{\pi}{6}\); sine is negative in Quadrants III and IV: \(\theta=\frac{7\pi}{6},\ \frac{11\pi}{6}\).

Scoring: 1 point for the reference angle and the two quadrants; 1 point for both angles (1 point if only one is given).

Question 7: Sine and tangent from cosine (2 points)

1. \(\sin^{2}\theta=1-\frac{144}{169}=\frac{25}{169}\); sine is positive in Quadrant II: \(\frac{5}{13}\).
2. \(\tan\theta=\frac{5/13}{-12/13}=\)\(-\frac{5}{12}\).

Scoring: 1 point for \(\sin\theta\) with the correct sign; 1 point for \(\tan\theta\).

Question 8: Using symmetry (3 points)

1. \(\cos^{2}\theta=1-\frac{16}{25}=\frac{9}{25}\); Quadrant III: \(\cos\theta=-\frac{3}{5}\).
2. \(\tan\theta=\frac{-4/5}{-3/5}=\)\(\frac{4}{3}\).
3. The angle \(-\theta\) reflects the point across the x-axis: \(\sin(-\theta)=\frac{4}{5}\) and \(\cos(-\theta)=-\frac{3}{5}\).

Scoring: 1 point per part; in part c, 1 point requires both values.

Question 9: Ferris wheel (3 points)

1. \(25+20\cdot\frac{1}{2}=\)35 ft.
2. \(25+20(-\frac{\sqrt{3}}{2})=25-10\sqrt{3}\approx \)7.68 ft.
3. \(20\sin\theta=-10\), so \(\sin\theta=-\frac{1}{2}\): \(\theta=\frac{7\pi}{6},\ \frac{11\pi}{6}\).

Scoring: 1 point per part.

⬇ Download the Algebra 2 Radians and the Unit Circle test answer key (PDF, 103 KB)

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