Find the exact value of each expression.
sin (-13π/12)

Verified step by step guidance

1. Start by recognizing that \(-\frac{13\pi}{12}\) is an angle in radians. Since it is negative, it indicates a clockwise rotation from the positive x-axis. To simplify, convert this angle to a positive angle by adding \(2\pi\) (a full circle) to it: \(-\frac{13\pi}{12} + 2\pi\).
2. Simplify the expression from Step 1 to find the equivalent positive angle. This will help in identifying the reference angle and the quadrant in which the angle lies.
3. Once you have the positive angle, determine the reference angle. The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.
4. Determine the quadrant in which the angle \(-\frac{13\pi}{12}\) lies. This is important because the sign of the sine function depends on the quadrant.
5. Use the reference angle and the quadrant information to find the sine of the original angle. Remember that the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants.

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 interpretation of the sine, cosine, and tangent functions. Angles measured in radians correspond to points on the unit circle, where the x-coordinate represents cosine and the y-coordinate represents sine.

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is used to simplify the calculation of trigonometric functions for angles greater than 90 degrees or negative angles. For example, to find sin(-13π/12), we can determine its reference angle by adding 2π until the angle is positive and then finding the sine of the corresponding acute angle.

Sine Function Properties

The sine function is periodic with a period of 2π, meaning sin(θ) = sin(θ + 2kπ) for any integer k. Additionally, sine is an odd function, which implies that sin(-θ) = -sin(θ). These properties allow us to evaluate sine for negative angles and angles outside the standard range, facilitating the calculation of sin(-13π/12) by using its positive equivalent.