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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Recall the formula for the cosine of the difference of two angles: \(\cos(A - B) = \cos A \cos B + \sin A \sin B\).

Identify the angles in the problem: \(A = 45^\circ\) and \(B = 30^\circ\).

Substitute the values into the formula: \(\cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ\).

Use known exact values for sine and cosine of special angles: \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), and \(\sin 30^\circ = \frac{1}{2}\).

Write the expression with these values substituted: \(\left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)\), then simplify step-by-step.

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주요 개념

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Cosine of the Difference of Two Angles Formula

This formula states that cos(A - B) = cos A cos B + sin A sin B. It allows you to find the cosine of the difference between two angles by using the cosines and sines of the individual angles, enabling exact value calculations without a calculator.

Exact Values of Common Angles

Certain angles like 30°, 45°, and 60° have well-known sine and cosine values expressed in simple radicals (e.g., cos 30° = √3/2, sin 45° = √2/2). Knowing these exact values is essential for computing trigonometric expressions precisely.

Trigonometric Function Properties and Identities

Understanding the basic properties of sine and cosine functions, such as their ranges and periodicity, helps in applying identities correctly. Recognizing how these functions behave ensures accurate substitution and simplification in problems.