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

Verified step by step guidance

1. Recognize that the angle is negative: \(-\frac{13\pi}{12}\). Use the identity \(\sin(-\theta) = -\sin(\theta)\) to rewrite the expression as \(-\sin\left(\frac{13\pi}{12}\right)\).
2. Express \(\frac{13\pi}{12}\) as a sum of angles whose sine and cosine values are known. For example, \(\frac{13\pi}{12} = \pi + \frac{\pi}{12}\) or \(\frac{13\pi}{12} = \frac{3\pi}{4} + \frac{\pi}{3}\).
3. Use the sine addition formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) to expand \(\sin\left(\frac{13\pi}{12}\right)\) based on the chosen angle decomposition.
4. Substitute the exact values of \(\sin\) and \(\cos\) for the known angles (like \(\frac{3\pi}{4}\) and \(\frac{\pi}{3}\)) into the expanded expression.
5. Simplify the expression algebraically to find the exact value of \(\sin\left(\frac{13\pi}{12}\right)\), then multiply by \(-1\) to get the value of \(\sin\left(-\frac{13\pi}{12}\right)\).

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 are measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles on the unit circle, including negative angles, is essential for evaluating trigonometric functions like sine.

Reference Angles and Angle Reduction

Reference angles help simplify the evaluation of trigonometric functions by relating any angle to an acute angle in the first quadrant. For angles outside the standard range, such as negative or large angles, reducing them by adding or subtracting multiples of 2π or using angle sum/difference identities is necessary.

Sine Function Properties and Identities

The sine function is periodic with period 2π and odd, meaning sin(-θ) = -sin(θ). Using these properties along with angle sum and difference formulas allows exact evaluation of sine for angles like -13π/12 by expressing them as sums or differences of known angles.