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

Verified step by step guidance

1. insert step 1> Convert the angle from degrees to radians if necessary, but since we are working with degrees, we can proceed with 240° directly.
2. insert step 2> Recognize that 240° is in the third quadrant of the unit circle.
3. insert step 3> Recall that in the third quadrant, the sine function is negative.
4. insert step 4> Determine the reference angle for 240° by subtracting 180° from 240°, which gives 60°.
5. insert step 5> Use the reference angle to find the sine value: \( \sin(240°) = -\sin(60°) \). Since \( \sin(60°) = \frac{\sqrt{3}}{2} \), the exact value of \( \sin(240°) \) is \( -\frac{\sqrt{3}}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured from the positive x-axis, allowing for easy calculation of trigonometric values for various angles.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For angles greater than 180°, like 240°, the reference angle helps determine the sine and cosine values by relating them to an angle in the first quadrant. In this case, the reference angle for 240° is 240° - 180° = 60°, which is crucial for finding the sine value.

Sine Function in Quadrants

The sine function varies in sign depending on the quadrant in which the angle lies. In the third quadrant, where 240° is located, the sine value is negative. Understanding the signs of sine, cosine, and tangent in each quadrant is essential for accurately determining the exact value of trigonometric functions without a calculator.