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

The first step of subtracting \(2\pi\) from \(\frac{10\pi}{3}\) to get \(\frac{4\pi}{3}\) is correct since \(\frac{10\pi}{3}-2\pi=\frac{10\pi - 6\pi}{3}=\frac{4\pi}{3}\).

Step2: Analyze step - 2

The error is in finding the reference - angle. The angle \(\frac{4\pi}{3}\) is in the third quadrant. The reference - angle \(\theta_{r}\) for an angle \(\theta=\frac{4\pi}{3}\) in the third quadrant is \(\theta-\pi\), so \(\frac{4\pi}{3}-\pi=\frac{4\pi - 3\pi}{3}=\frac{\pi}{3}\), not \(\frac{3\pi}{2}-\frac{4\pi}{3}=\frac{\pi}{6}\).

Step3: Analyze step - 3

Since the reference - angle was calculated wrong in step 2, the cosine value used for the reference - angle is incorrect. The cosine of the correct reference - angle \(\frac{\pi}{3}\) is \(\frac{1}{2}\), not \(\frac{\sqrt{3}}{2}\) (which is the cosine of \(\frac{\pi}{6}\)).

Step4: Analyze step - 4

The angle \(\frac{4\pi}{3}\) is in the third quadrant, and in the third quadrant, \(\cos\theta<0\), not positive as stated in step 4.

Answer:

Henry made errors in calculating the reference - angle (step 2), using the wrong cosine value for the reference - angle (step 3), and misidentifying the sign of the cosine function (step 4).