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Chapter 4: Problem 66

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$\frac{10 \pi}{3}$$

Short Answer

Expert verified
The sine, cosine and tangent of the angle \(\frac{10 \pi}{3}\) are \(- \frac{1}{2}\), \(- \frac{\sqrt{3}}{2}\), and \(\sqrt{3}\) respectively.

Step by step solution

01

Decode the Given Angle

The angle given is \(\frac{10 \pi}{3}\). To find the position of this angle on the unit circle, we recognize that \(\frac{10 \pi}{3}\) is more than \(2 \pi\). So, it means it has circled around once and then went on \( \frac{4 \pi}{3}\) further.
02

Determine the Quadran of the Angle

\(\frac{4 \pi}{3}\) corresponds to \(\frac{2\pi}{3}\) beyond \(\pi\), as a result, it is in the third quadrant.
03

Calculate the sine, cosine and tangent

In the third quadrant, sine and cosine are negative while tangent is positive. Since \(\frac{4 \pi}{3}\) is equivalent to \(\frac{2 \pi}{3}\) in magnitude (only differing by their directions), we can safely say that \(\frac{2 \pi}{3}\) is just the reflection of angle \(\frac{\pi}{3}\) (which is in first quadrant) in third quadrant. And the sine, cosine and tangent values of \(\frac{\pi}{3}\) are \(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\) and \(\sqrt{3}\) respectively.
04

Apply the signs to the values

In the third quadrant, sine and cosine are negative while tangent is positive. So the sine, cosine and tangent of \(\frac{10 \pi}{3}\) will be \(- \frac{1}{2}\), \(- \frac{\sqrt{3}}{2}\), and \(\sqrt{3}\) respectively.

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