Find the exact value of each expression.
sin 255°

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Recognize that 255° is an angle in the third quadrant where sine values are negative, but first express 255° as a sum or difference of special angles whose sine and cosine values are known. For example, write 255° as 180° + 75°.

Use the sine angle addition formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). Here, let \(a = 180^\circ\) and \(b = 75^\circ\).

Substitute into the formula: \(\sin 255^\circ = \sin 180^\circ \cos 75^\circ + \cos 180^\circ \sin 75^\circ\).

Recall the exact values: \(\sin 180^\circ = 0\), \(\cos 180^\circ = -1\), and use known exact values or formulas for \(\sin 75^\circ\) and \(\cos 75^\circ\) (which can be found using angle sum identities such as \(75^\circ = 45^\circ + 30^\circ\)).

Simplify the expression by substituting these values and combining terms to find the exact value of \(\sin 255^\circ\).

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Key Concepts

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

Reference Angles

A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps simplify trigonometric calculations by relating angles greater than 90° to their acute counterparts, allowing the use of known sine, cosine, or tangent values.

Unit Circle and Angle Quadrants

The unit circle divides angles into four quadrants, each with specific signs for sine and cosine values. Knowing that 255° lies in the third quadrant, where sine is negative, helps determine the sign of the trigonometric function's value.

Angle Sum and Difference Identities

These identities express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For example, sin(255°) can be rewritten as sin(180° + 75°), allowing the use of known sine and cosine values to find the exact value.

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