Find the exact value of each expression.
sin (- 5π/12)

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Recognize that the angle is negative: \(-\frac{5\pi}{12}\). Use the identity \(\sin(-\theta) = -\sin(\theta)\) to rewrite the expression as \(-\sin\left(\frac{5\pi}{12}\right)\).

Express \(\frac{5\pi}{12}\) as a sum of angles whose sine and cosine values are known. For example, \(\frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}\).

Use the sine addition formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). Substitute \(a = \frac{\pi}{3}\) and \(b = \frac{\pi}{4}\) to get \(\sin\left(\frac{5\pi}{12}\right) = \sin\frac{\pi}{3} \cos\frac{\pi}{4} + \cos\frac{\pi}{3} \sin\frac{\pi}{4}\).

Recall the exact values: \(\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\), \(\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{3} = \frac{1}{2}\), and \(\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\). Substitute these into the expression.

Combine the terms and simplify the expression inside the parentheses, then apply the negative sign from step 1 to find the exact value of \(\sin\left(-\frac{5\pi}{12}\right)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 which represent clockwise rotation, is essential for evaluating trigonometric functions like sine.

Sine Function and Its Properties

The sine function relates an angle to the y-coordinate of the corresponding point on the unit circle. It is an odd function, meaning sin(-θ) = -sin(θ), which helps simplify expressions involving negative angles. Knowing this property allows for easier calculation of sine values for negative angles.

Angle Sum and Difference Identities

These identities express the sine of sums or differences of angles in terms of sines and cosines of individual angles. For example, sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). They are useful for finding exact values of angles like 5π/12 by breaking them into sums or differences of special angles with known sine and cosine values.

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