Algebra 2 Radians and the Unit Circle Test Solutions and Scoring (PDF)

1. \(150 \cdot \frac{\pi}{180} = \frac{5\pi}{6}\)

2. \(\frac{5\pi}{3} \cdot \frac{180}{\pi} = 300\)

(a) \(\frac{5\pi}{6}\); (b) \(300^{\circ}\)

Scoring: 1 point for each conversion.

\(s = 8 \cdot \frac{5\pi}{6} = \frac{20\pi}{3}\).

20π/3 cm

Scoring: 1 point for s = rθ, 1 point for the value.

1. \(\frac{\sqrt{3}}{2}\)

2. \(\frac{\sqrt{2}}{2}\)

3. \(\frac{\sqrt{3}}{3}\)

(a) \(\frac{\sqrt{3}}{2}\); (b) \(\frac{\sqrt{2}}{2}\); (c) \(\frac{\sqrt{3}}{3}\)

Scoring: 1 point for each value.

1. Quadrant III, reference \(\frac{\pi}{3}\): cosine negative, \(- \frac{1}{2}\).

2. Quadrant IV, reference \(\frac{\pi}{6}\): sine negative, \(- \frac{1}{2}\).

3. Quadrant III, reference \(\frac{\pi}{4}\): tangent positive, \(1\).

(a) \(- \frac{1}{2}\); (b) \(- \frac{1}{2}\); (c) \(1\)

Scoring: 1 point for each value.

Reference angle \(\frac{\pi}{4}\); cosine is positive in quadrants I and IV.

\(\theta = \frac{\pi}{4}\) or \(2\pi – \frac{\pi}{4} = \frac{7\pi}{4}\).

\(\theta = \frac{\pi}{4}, \frac{7\pi}{4}\)

Scoring: 1 point for each solution.

\(\frac{9\pi}{4} – 2\pi = \frac{\pi}{4}\), so \(\sin\frac{9\pi}{4} = \frac{\sqrt{2}}{2}\).

\(-\frac{2\pi}{3}\) is in quadrant III with reference angle \(\frac{\pi}{3}\): \(-\frac{1}{2}\).

\(\frac{\sqrt{2}}{2}\) and \(-\frac{1}{2}\)

Scoring: 1 point for each value.

Reference angle \(\frac{\pi}{4}\), quadrant II: x negative, y positive.

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

Scoring: 1 point for each coordinate.

\(s = 14 \cdot \frac{5\pi}{2} = 35\pi\) inches, about 110 inches.

35π in, about 110 in

Scoring: 1 point for s = rθ, 1 point for the value.

\(\tan\frac{\pi}{2} = \frac{\sin(\pi/2)}{\cos(\pi/2)} = \frac{1}{0}\), which is undefined. A zero in the denominator makes tangent undefined, not zero.

tan(π/2) is undefined

Scoring: 1 point for the ratio, 1 point for the conclusion.