Use a sum or difference formula to find the exact value of each expression. cos(45° + 30°)

Verified step by step guidance

1. Identify the formula to use: Since the expression is \( \cos(45^\circ + 30^\circ) \), we use the cosine sum formula, which is \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).
2. Assign the angles: Let \( A = 45^\circ \) and \( B = 30^\circ \).
3. Write the expression using the formula: \( \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \).
4. Recall the exact values of the trigonometric functions: \( \cos 45^\circ = \frac{\sqrt{2}}{2} \), \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), and \( \sin 30^\circ = \frac{1}{2} \).
5. Substitute these values into the expression and simplify step-by-step to find the exact value.

Key Concepts

Sum and Difference Formulas for Cosine

These formulas allow you to find the cosine of the sum or difference of two angles. Specifically, cos(A + B) = cos A cos B - sin A sin B, which helps compute exact values without a calculator.

Exact Values of Common Angles

Knowing the exact sine and cosine values for standard angles like 30° and 45° is essential. For example, cos 30° = √3/2 and sin 45° = √2/2, which are used directly in the sum formula.

Trigonometric Function Properties

Understanding the behavior and relationships of sine and cosine functions, such as their ranges and symmetry, aids in correctly applying formulas and simplifying results.