Question

henry is asked to find the exact value of cos(10π/3). his steps are shown below. 1. subtract 2π from 10π/3 as many times as possible: 10π/3 - 2π = 4π/3. 2. find the reference angle for 4π/3: 3π/2 - 4π/3 = π/6. 3. the cosine value for π/6 is √3/2. 4. the cosine value is positive because π/6 is in the first quadrant. which of the following describes henrys errors?

Explanation:

Step1: Analyze step - 1

Subtracting \(2\pi\) from \(\frac{10\pi}{3}\) to get a co - terminal angle is correct. \(\frac{10\pi}{3}-2\pi=\frac{10\pi - 6\pi}{3}=\frac{4\pi}{3}\).

Step2: Analyze step - 2

The reference angle for an angle \(\theta=\frac{4\pi}{3}\) in the third quadrant is found by \(\theta-\pi\). The correct reference - angle calculation for \(\frac{4\pi}{3}\) is \(\frac{4\pi}{3}-\pi=\frac{\pi}{3}\), not \(\frac{3\pi}{2}-\frac{4\pi}{3}\).

Step3: Analyze step - 3

Since the reference angle is \(\frac{\pi}{3}\), the cosine of the reference angle \(\cos(\frac{\pi}{3})=\frac{1}{2}\).

Step4: Analyze step - 4

The angle \(\frac{4\pi}{3}\) is in the third quadrant where cosine is negative.

Answer:

Henry made an error in calculating the reference angle (step 2) and also in determining the sign of the cosine value (step 4). The correct reference angle for \(\frac{4\pi}{3}\) is \(\frac{\pi}{3}\) and \(\cos(\frac{10\pi}{3})=\cos(\frac{4\pi}{3})=-\frac{1}{2}\) because \(\frac{4\pi}{3}\) is in the third quadrant where cosine is negative.