In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. cos 50° cos 20° + sin 50° sin 20°

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Recognize that the given expression cos 50° cos 20° + sin 50° sin 20° matches the formula for \( \cos(\alpha - \beta) \), which is \( \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \).

Identify \( \alpha = 50° \) and \( \beta = 20° \) from the expression.

Substitute \( \alpha \) and \( \beta \) into the formula \( \cos(\alpha - \beta) \) to get \( \cos(50° - 20°) \).

Simplify the expression \( \cos(50° - 20°) \) to \( \cos(30°) \).

Use the known exact value of \( \cos(30°) \) to find the result.

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주요 개념

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Cosine of Angle Difference Formula

The cosine of the difference of two angles, α and β, is given by the formula cos(α - β) = cos(α)cos(β) + sin(α)sin(β). This formula is essential for simplifying expressions involving the cosine of angle differences and is widely used in trigonometric calculations.

Trigonometric Values

Understanding the exact values of trigonometric functions for common angles (like 0°, 30°, 45°, 60°, and 90°) is crucial. In this problem, knowing the values of cos(50°), cos(20°), sin(50°), and sin(20°) allows for the direct application of the cosine difference formula to find the exact value of the expression.

Exact Value Calculation

Finding the exact value of trigonometric expressions often involves substituting known values into formulas and performing arithmetic operations. In this case, substituting the values into the cosine of angle difference formula will yield the exact value of cos(50° - 20°), which simplifies to cos(30°).