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Chapter 4: Problem 13

Sketch each angle in standard position. $$\text { (a) } \frac{\pi}{3} \quad \text { (b) }-\frac{2 \pi}{3}$$

Short Answer

Expert verified
The angle \(\frac{\pi}{3}\) measures 60 degrees in counterclockwise direction, whereas the angle \(- \frac{2 \pi}{3}\) measures 120 degrees in the clockwise direction from the \(X\)-axis.

Step by step solution

01

Sketch the basic unit circle and axes

Draw a circle with its center at the origin. Mark the center as \(O\). Draw two evenly divided axes (X and Y-axis) passing through the center of the circle. Be sure to indicate the positive directions. The \(X\)-axis extending towards the right is the initial side for the angles.
02

Plot the first angle

\(\frac{\pi}{3}\) is 60 degrees measured in a counterclockwise direction from the initial side. Find the point at a 60-degree angle from the positive \(X\)-axis, lying on the circumference of the circle. Draw the angle by connecting this point with the origin and the point on the positive \(X\)-axis.
03

Plot the second angle

\(- \frac{2 \pi}{3}\) is -120 degrees. Negative angles are measured clockwise from the initial side. Find the point that's at a 120-degree angle measured clockwise from the \(X\)-axis, lying on the circle. Draw the angle by connecting this point with the origin and the point on the positive \(X\)-axis.

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