Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression.cos(45° - 30°)

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1. Identify the formula for the cosine of the difference of two angles: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).
2. Substitute the given angles into the formula: \( A = 45^\circ \) and \( B = 30^\circ \).
3. Calculate \( \cos 45^\circ \) and \( \cos 30^\circ \) using known values: \( \cos 45^\circ = \frac{\sqrt{2}}{2} \) and \( \cos 30^\circ = \frac{\sqrt{3}}{2} \).
4. Calculate \( \sin 45^\circ \) and \( \sin 30^\circ \) using known values: \( \sin 45^\circ = \frac{\sqrt{2}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \).
5. Substitute these values into the formula: \( \cos(45^\circ - 30^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \).

Key Concepts

Cosine of the Difference of Two Angles

The cosine of the difference of two angles is given by the formula cos(A - B) = cos(A)cos(B) + sin(A)sin(B). This identity allows us to express the cosine of the difference between two angles in terms of the cosines and sines of the individual angles, facilitating the calculation of exact values for trigonometric expressions.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions for common angles (like 0°, 30°, 45°, 60°, and 90°) are often memorized or derived from the unit circle. For instance, cos(45°) = √2/2 and cos(30°) = √3/2. Knowing these values is essential for solving problems involving trigonometric identities and expressions.

Angle Measurement in Degrees

In trigonometry, angles can be measured in degrees or radians. The question uses degrees, where a full circle is 360°. Understanding how to convert between degrees and radians, and how to work with angles in degrees, is crucial for applying trigonometric identities and solving related problems accurately.