In Exercises 57–64, find the exact value of the following under the given conditions:
b. sin (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

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Identify the given information: \(\cos \alpha = \frac{8}{17}\) with \(\alpha\) in quadrant IV, and \(\sin \beta = -\frac{1}{2}\) with \(\beta\) in quadrant III.

Determine \(\sin \alpha\) using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Since \(\cos \alpha = \frac{8}{17}\), calculate \(\sin \alpha = \pm \sqrt{1 - \left(\frac{8}{17}\right)^2}\). Because \(\alpha\) is in quadrant IV, where sine is negative, choose the negative root.

Determine \(\cos \beta\) using the Pythagorean identity \(\sin^2 \beta + \cos^2 \beta = 1\). Since \(\sin \beta = -\frac{1}{2}\), calculate \(\cos \beta = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2}\). Because \(\beta\) is in quadrant III, where cosine is negative, choose the negative root.

Use the angle sum identity for sine: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).

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

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주요 개념

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Trigonometric Angle Sum Identity

The sine of a sum of two angles, sin(α + β), can be found using the identity sin(α + β) = sin α cos β + cos α sin β. This formula allows the expression of the sine of a combined angle in terms of the sines and cosines of the individual angles.

Determining Signs of Trigonometric Functions by Quadrant

The sign of sine and cosine values depends on the quadrant in which the angle lies. In quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. This knowledge is essential to correctly assign signs when calculating unknown trigonometric values.

Using Pythagorean Identity to Find Missing Values

When one trigonometric value is given, the other can be found using the Pythagorean identity sin²θ + cos²θ = 1. For example, if cos α is known, sin α can be found by sin α = ±√(1 - cos²α), with the sign determined by the quadrant of α.

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.