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

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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 formula for the sine of a sum: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).

Find \(\cos \alpha\) using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since \(\alpha\) is in quadrant I, \(\cos \alpha\) is positive. Calculate \(\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{3}{5}\right)^2}\).

Find \(\cos \beta\) similarly using \(\cos \beta = \pm \sqrt{1 - \sin^2 \beta} = \pm \sqrt{1 - \left(\frac{5}{13}\right)^2}\). Since \(\beta\) is in quadrant II, \(\cos \beta\) is negative.

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

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주요 개념

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Sine Addition Formula

The sine addition formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine 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 calculation.

Quadrant Sign Rules for Trigonometric Functions

The signs of sine and cosine functions vary by quadrant: in quadrant I, both sine and cosine are positive; in quadrant II, sine is positive and cosine is negative. Correctly applying these sign rules is crucial to accurately compute sin(α + β).

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