In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin 240°

Verified step by step guidance

1. Recognize that the angle 240° is in the third quadrant of the unit circle, where sine values are negative.
2. Find the reference angle for 240° by subtracting 180°: \(240^\circ - 180^\circ = 60^\circ\).
3. Recall the sine value of the reference angle 60°, which is \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).
4. Since 240° is in the third quadrant where sine is negative, apply the sign: \(\sin 240^\circ = -\sin 60^\circ\).
5. Write the exact value as \(\sin 240^\circ = -\frac{\sqrt{3}}{2}\).

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles in trigonometry are often measured in degrees or radians, and their position on the unit circle determines the values of sine and cosine. Understanding how to locate 240° on the unit circle is essential for finding sin 240°.

Reference Angles

A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. For angles greater than 180°, like 240°, the reference angle helps simplify the calculation of sine and cosine by relating them to known values of acute angles.

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant in which the angle lies. Since 240° is in the third quadrant, where sine values are negative, this knowledge helps determine the correct sign of sin 240° after using the reference angle.