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Chapter 4: Problem 59

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$\frac{2 \pi}{3}$$

Short Answer

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

Step by step solution

01

Identifying the angle on the unit circle

To solve this, let's first understand where \(\frac {2 \pi}{3}\) appears on the unit circle. As we know, \(\pi\) radians is equivalent to 180 degree. Therefore, \(\frac {2 \pi}{3}\) = \(\frac {2*180}{3} = 120^{\circ}\). So, this angle is in the second quadrant.
02

Evaluating Sine value

The sine of an angle in the unit circle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. In the second quadrant, sine is positive. Thus, \(\sin \frac {2 \pi}{3}\) = \(\sin 120^{\circ} = \sqrt {3}/2\)
03

Evaluating Cosine value

The cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. In the second quadrant, cosine is negative. Thus, \(\cos \frac {2 \pi}{3}\) = \(\cos 120^{\circ} = -1/2\)
04

Evaluating Tangent value

The tangent of an angle is the sine of that angle divided by the cosine of that angle. So, \(\tan \frac {2 \pi}{3}\) = \(\frac {\sin \frac {2 \pi}{3}}{\cos \frac {2 \pi}{3}} = \frac {\sqrt {3}/2}{-1/2} = - \sqrt {3}\)

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