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 75°

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1. Recognize that 75° can be expressed as the sum of two special angles whose sine and cosine values are well known. For example, 75° = 45° + 30°.
2. Recall the sine sum identity: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\).
3. Substitute \(A = 45^\circ\) and \(B = 30^\circ\) into the identity to get \(\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\).
4. Use the exact values of sine and cosine for 30° and 45°: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\sin 30^\circ = \frac{1}{2}\), and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
5. Substitute these values into the expression and simplify the resulting expression to find the exact value of \(\sin 75^\circ\).

Key Concepts

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 the exact evaluation of trigonometric functions at angles not commonly found on the unit circle.

Exact Values of Common Angles

Certain angles like 30°, 45°, and 60° have well-known exact sine and cosine values derived from special triangles. Knowing these values is essential for applying sum and difference identities to find exact values of other angles, such as 75°, by expressing them as sums or differences of these common angles.

Evaluating Trigonometric Expressions

To find the exact value of expressions like sin 75°, one must substitute the sum or difference of angles into the identity, then replace sine and cosine of the known angles with their exact values. Simplifying the resulting expression yields the exact trigonometric value without using a calculator.