In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

검증된 단계별 안내

Identify the given information: \(\sin \alpha = \frac{3}{5}\) with \(\alpha\) in quadrant I, and \(\sin \beta = \frac{5}{13}\) with \(\beta\) in quadrant II.

Recall the Pythagorean identity to find \(\cos \alpha\) and \(\cos \beta\). Since \(\sin^2 \theta + \cos^2 \theta = 1\), calculate \(\cos \alpha = \sqrt{1 - \sin^2 \alpha}\) and \(\cos \beta = \pm \sqrt{1 - \sin^2 \beta}\).

Determine the correct signs for \(\cos \alpha\) and \(\cos \beta\) based on the quadrant information: in quadrant I, cosine is positive; in quadrant II, cosine is negative.

Use the cosine addition formula: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\).

Substitute the values of \(\sin \alpha\), \(\cos \alpha\), \(\sin \beta\), and \(\cos \beta\) into the formula and simplify to find the exact value of \(\cos(\alpha + \beta)\).

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주요 개념

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Sum of Angles Formula for Cosine

The sum of angles formula states that cos(α + β) = cos α cos β - sin α sin β. This identity allows us to find the cosine of the sum of two angles using the sines and cosines of the individual angles, which is essential for solving the problem.

Determining Cosine from Sine and Quadrant

Given sin α and sin β along with their quadrants, we use the Pythagorean identity cos²θ = 1 - sin²θ to find cos α and cos β. The sign of cosine depends on the quadrant: positive in quadrant I and negative in quadrant II, which affects the final value.

Quadrant Sign Rules for Trigonometric Functions

The sign of sine and cosine functions depends on the quadrant of the angle. In quadrant I, both sine and cosine are positive; in quadrant II, sine is positive but cosine is negative. Correctly applying these sign rules is crucial for accurate calculation.

In Exercises 57–64, find the exact value of the following under the given conditions: a. cos (α + β) sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.