henry is asked to find the exact value of cos(10π/3). his steps are shown below. 1. subtract 2π from 10π/3…

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).