In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (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 cosine addition formula: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\).

Substitute the values of \(\cos \alpha\), \(\cos \beta\), \(\sin \alpha\), and \(\sin \beta\) 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 Sine and Cosine Values from Quadrants

Knowing the quadrant in which an angle lies helps determine the sign of its sine and cosine values. For example, in quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. This is crucial for correctly assigning signs to trigonometric values.

Using Pythagorean Identity to Find Missing Values

The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of the missing sine or cosine value when one is known. This is important here to find sin α and cos β from the given values, enabling the use of the sum formula accurately.

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