Use the following conditions to solve Exercises 1–4: 4 𝝅 sin α = ----- , ------- \u003c α \u003c 𝝅 5 2 5 𝝅 cos β = ------ , 0 \u003c β \u003c ------ 13 2 Find the exact value of each of the following. cos (α + β)

Ch. 4 - Laws of Sines and Cosines; Vectors

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Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

문제 1

4장, 문제 1

Use the following conditions to solve Exercises 1–4: 4 𝝅 sin α = ----- , ------- < α < 𝝅 5 2 5 𝝅 cos β = ------ , 0 < β < ------ 13 2 Find the exact value of each of the following. cos (α + β)

검증된 단계별 안내

Identify the formula for the cosine of the sum of two angles: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).

Determine \( \cos \alpha \) using the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Since \( \sin \alpha = \frac{4\pi}{5} \), calculate \( \cos \alpha \).

Determine \( \sin \beta \) using the Pythagorean identity: \( \sin^2 \beta + \cos^2 \beta = 1 \). Since \( \cos \beta = \frac{5\pi}{13} \), calculate \( \sin \beta \).

Substitute the values of \( \cos \alpha \), \( \cos \beta \), \( \sin \alpha \), and \( \sin \beta \) into the formula for \( \cos(\alpha + \beta) \).

Simplify the expression to find the exact value of \( \cos(\alpha + \beta) \).

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주요 개념

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Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. The sine function, sin(θ), represents the ratio of the length of the opposite side to the hypotenuse, while the cosine function, cos(θ), represents the ratio of the adjacent side to the hypotenuse. Understanding these functions is crucial for solving problems involving angles and their relationships.

Angle Addition Formula

The angle addition formulas for sine and cosine allow us to find the sine or cosine of the sum of two angles. Specifically, cos(α + β) = cos(α)cos(β) - sin(α)sin(β). This formula is essential for calculating the cosine of the sum of angles when the individual angles are known, as in the given problem.

Quadrants and Angle Ranges

The range of angles specified in the problem indicates which quadrant the angles α and β lie in. For instance, the range for α is between 0 and π, placing it in the first or second quadrant, while β is between 0 and π/2, placing it in the first quadrant. Understanding the signs of sine and cosine in different quadrants is vital for determining the correct values of these functions when solving trigonometric equations.