In Exercises 8–12, draw each angle in standard position. 8𝜋 3

Verified step by step guidance

1. Understand that an angle in standard position is drawn with its vertex at the origin and its initial side along the positive x-axis.
2. Identify the given angle: \(\frac{8\pi}{3}\). Notice that this angle is greater than \(2\pi\), which means it makes more than one full rotation.
3. To find the equivalent angle between \(0\) and \(2\pi\), subtract multiples of \(2\pi\) from \(\frac{8\pi}{3}\) until the result lies within one full rotation: calculate \(\frac{8\pi}{3} - 2\pi\).
4. Simplify the subtraction: express \(2\pi\) as \(\frac{6\pi}{3}\) to get \(\frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3}\).
5. Draw the angle \(\frac{2\pi}{3}\) in standard position by starting from the positive x-axis and rotating counterclockwise by \(\frac{2\pi}{3}\) radians.

Key Concepts

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The angle is measured by rotating the initial side to the terminal side, either counterclockwise for positive angles or clockwise for negative angles.

Radian Measure

Radian measure defines angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Since 2π radians equal 360 degrees, angles can be converted between radians and degrees for easier interpretation and drawing.

Coterminal Angles

Coterminal angles share the same terminal side but differ by full rotations of 2π radians. To draw an angle like 8π/3, it is often helpful to subtract multiples of 2π to find a coterminal angle between 0 and 2π, simplifying the drawing process.