Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value.sin 75° + sin 15°

Verified step by step guidance

1. Identify the sum-to-product identity for sine: \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \).
2. Assign \( A = 75^\circ \) and \( B = 15^\circ \).
3. Calculate \( \frac{A + B}{2} = \frac{75^\circ + 15^\circ}{2} = 45^\circ \).
4. Calculate \( \frac{A - B}{2} = \frac{75^\circ - 15^\circ}{2} = 30^\circ \).
5. Substitute these values into the identity: \( 2 \sin(45^\circ) \cos(30^\circ) \).

Key Concepts

Sum-to-Product Formulas

Sum-to-product formulas are trigonometric identities that express the sum or difference of two sine or cosine functions as a product. For example, the formula for the sum of sines states that sin(A) + sin(B) can be rewritten as 2 sin((A+B)/2) cos((A-B)/2). These formulas simplify calculations and are essential for transforming expressions in trigonometry.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent for commonly used angles, such as 0°, 30°, 45°, 60°, and 90°. Knowing these values allows for quick calculations and simplifications in trigonometric problems. For instance, sin(30°) = 1/2 and cos(60°) = 1/2 are exact values that can be used in various trigonometric identities.

Angle Addition and Subtraction

Angle addition and subtraction formulas are used to find the sine, cosine, or tangent of the sum or difference of two angles. For example, sin(A + B) = sin(A)cos(B) + cos(A)sin(B) and sin(A - B) = sin(A)cos(B) - cos(A)sin(B). These formulas are crucial for breaking down complex angles into simpler components, facilitating easier calculations in trigonometric expressions.