In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

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Recognize that the expression \(2 \sin 15^\circ \cos 15^\circ\) matches the double-angle identity for sine, which is \(2 \sin A \cos A = \sin 2A\).

Rewrite the expression using the identity: \(2 \sin 15^\circ \cos 15^\circ = \sin (2 \times 15^\circ)\).

Simplify the angle inside the sine function: \(\sin (30^\circ)\).

Recall the exact value of \(\sin 30^\circ\), which is a commonly known special angle in trigonometry.

Conclude that the exact value of the original expression is the exact value of \(\sin 30^\circ\).

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Double-Angle Identities

Double-angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For sine, the identity is sin(2θ) = 2 sin θ cos θ, which allows rewriting products like 2 sin 15° cos 15° as sin 30°.

Evaluating Exact Trigonometric Values

Exact values of trigonometric functions at special angles (like 30°, 45°, 60°) are well-known and can be used to find precise results without a calculator. For example, sin 30° equals 1/2, which helps in determining the exact value of the expression.

Trigonometric Function Notation and Simplification

Understanding how to rewrite expressions using trigonometric notation and simplify them is essential. Recognizing patterns such as products of sine and cosine that match double-angle formulas enables efficient simplification and evaluation.