Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

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Identify the sum or difference identity that applies to the expression. Since the expression is \( \sin(60^\circ - 45^\circ) \), use the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).

Substitute \( A = 60^\circ \) and \( B = 45^\circ \) into the identity: \( \sin(60^\circ - 45^\circ) = \sin 60^\circ \cos 45^\circ - \cos 60^\circ \sin 45^\circ \).

Recall the exact values of the sine and cosine for the special angles: \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), \( \cos 60^\circ = \frac{1}{2} \), \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), and \( \cos 45^\circ = \frac{\sqrt{2}}{2} \).

Replace the trigonometric functions in the expression with their exact values: \( \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) - \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) \).

Simplify the expression by multiplying the fractions and combining like terms to find the exact value.

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Sum and Difference Identities

Sum and difference identities express the sine, cosine, or tangent of a sum or difference of two angles in terms of the sines and cosines of the individual angles. For sine, the identity is sin(a - b) = sin(a)cos(b) - cos(a)sin(b), which allows exact evaluation of expressions like sin(60° - 45°).

Exact Values of Common Angles

Certain angles such as 30°, 45°, 60°, and their multiples have known exact sine and cosine values involving square roots. For example, sin(45°) = √2/2 and cos(60°) = 1/2. These values are essential for calculating exact trigonometric expressions without a calculator.

Angle Measurement in Degrees

Understanding that the angles given are in degrees is crucial for applying the correct trigonometric values and identities. Degrees are a common unit for measuring angles, and knowing how to work with degree measures ensures proper use of trigonometric formulas and exact values.