Find the exact value of each expression. See Example 3. sin 1305°

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the sine, cosine, and tangent functions based on the coordinates of points on a circle with a radius of one. Angles are measured from the positive x-axis, and the coordinates of any point on the circle correspond to the cosine and sine of that angle. Understanding the unit circle helps in determining the values of trigonometric functions for any angle, including those greater than 360°.

Angle Reduction

Angle reduction is a technique used to simplify angles that exceed 360° or are negative by finding an equivalent angle within the standard range of 0° to 360°. This is achieved by subtracting or adding multiples of 360° to the original angle. For example, to find sin 1305°, one would reduce it by subtracting 360° multiple times until the angle falls within the standard range, making it easier to evaluate the sine function.

Sine Function Properties

The sine function is periodic with a period of 360°, meaning that sin(θ) = sin(θ + 360°n) for any integer n. Additionally, sine is positive in the first and second quadrants and negative in the third and fourth quadrants. Understanding these properties allows for the determination of sine values for various angles, including those that are coterminal with the original angle, which is essential for finding the exact value of sin 1305°.