To find the exact value of [tex]\(\cos \frac{10 \pi}{3}\)[/tex], let's proceed through the problem step by step and identify any errors in Henry's process.
1. Normalize the angle:
To get an angle within the range of [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex], we perform:
[tex]\[
\frac{10 \pi}{3} - 2 \pi = \frac{10 \pi}{3} - \frac{6 \pi}{3} = \frac{4 \pi}{3}
\][/tex]
This is correct. The normalized angle is [tex]\(\frac{4 \pi}{3}\)[/tex].
2. Determine the reference angle:
The angle [tex]\(\frac{4 \pi}{3}\)[/tex] is located in the third quadrant. To find the reference angle, subtract [tex]\(\pi\)[/tex] from [tex]\(\frac{4 \pi}{3}\)[/tex]:
[tex]\[
\text{Reference angle} = \frac{4 \pi}{3} - \pi = \frac{4 \pi}{3} - \frac{3 \pi}{3} = \frac{\pi}{3}
\][/tex]
The reference angle is [tex]\(\frac{\pi}{3}\)[/tex], not [tex]\(\frac{\pi}{6}\)[/tex]. Henry made an error in finding the reference angle.
3. Cosine of the reference angle:
The cosine value for the reference angle [tex]\(\frac{\pi}{3}\)[/tex] is:
[tex]\[
\cos \frac{\pi}{3} = \frac{1}{2}
\][/tex]
This is correct.
4. Determine the cosine value considering the quadrant:
Since [tex]\(\frac{4 \pi}{3}\)[/tex] is in the third quadrant, where cosine values are negative, the cosine value becomes:
[tex]\[
\cos \frac{4 \pi}{3} = -\cos \frac{\pi}{3} = -\frac{1}{2}
\][/tex]
This is also correct.
Therefore, summarizing Henry's errors:
- In step 2, Henry incorrectly determined the reference angle for [tex]\(\frac{4 \pi}{3}\)[/tex]. The correct reference angle is [tex]\(\frac{\pi}{3}\)[/tex], not [tex]\(\frac{\pi}{6}\)[/tex].
- In step 3, Henry incorrectly used the value for the cosine of [tex]\(\frac{\pi}{6}\)[/tex], but correctly stating [tex]\(\cos \frac{\pi}{3}\)[/tex] would have been [tex]\(\frac{1}{2}\)[/tex].
- Additionally, in step 4, he incorrectly stated that the cosine value was positive because he used the reference angle out of context with the correct quadrant. The cosine value should be negative in the third quadrant.
Putting this all together, the correct solution is:
[tex]\[
\cos \frac{10 \pi}{3} = -\frac{1}{2}
\][/tex]
