In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.
29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)

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Identify the given expression: \(\sin \frac{5\pi}{12} \cos \frac{\pi}{4} - \cos \frac{5\pi}{12} \sin \frac{\pi}{4}\).

Recognize that this expression matches the sine difference identity: \(\sin A \cos B - \cos A \sin B = \sin(A - B)\).

Assign \(A = \frac{5\pi}{12}\) and \(B = \frac{\pi}{4}\), then rewrite the expression as \(\sin\left( \frac{5\pi}{12} - \frac{\pi}{4} \right)\).

Calculate the difference inside the sine function: find a common denominator and subtract \(\frac{5\pi}{12} - \frac{\pi}{4}\).

Express the original expression as \(\sin\) of the resulting angle and then evaluate the exact value using known sine values or special angle properties.

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주요 개념

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Sum and Difference Identities

Sum and difference identities express the sine, cosine, or tangent of sums or differences of angles in terms of the sines and cosines of the individual angles. For example, sin(a ± b) = sin a cos b ± cos a sin b. These identities allow rewriting complex trigonometric expressions as single trigonometric functions of combined angles.

Exact Values of Trigonometric Functions

Certain angles, especially multiples of π/6, π/4, and π/3, have known exact sine, cosine, and tangent values. Using these exact values avoids decimal approximations and provides precise results. Recognizing these angles helps in evaluating expressions after applying sum or difference identities.

Angle Conversion and Simplification

Understanding how to manipulate and simplify angles, such as converting fractions of π into recognizable standard angles, is essential. This includes reducing angles to their reference angles within the unit circle to apply known values and identities effectively, facilitating the evaluation of trigonometric expressions.

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)