In Exercises 35–38, find the exact value of the following under the given conditions:
b. cos(α﹣β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

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Identify the given information and the intervals for \( \alpha \) and \( \beta \): - \( \sin \alpha = -\frac{1}{3} \) with \( \pi < \alpha < \frac{3\pi}{2} \) - \( \cos \beta = -\frac{1}{2} \) with \( \pi < \beta < \frac{3\pi}{2} \) These intervals indicate that both angles are in the third quadrant.

Recall the formula for \( \cos(\alpha - \beta) \): \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

Since \( \sin \alpha \) is given, find \( \cos \alpha \) using the Pythagorean identity: \[ \cos^2 \alpha = 1 - \sin^2 \alpha \] Calculate \( \cos \alpha \) considering the quadrant (third quadrant means \( \cos \alpha < 0 \)).

Similarly, since \( \cos \beta \) is given, find \( \sin \beta \) using the Pythagorean identity: \[ \sin^2 \beta = 1 - \cos^2 \beta \] Determine the sign of \( \sin \beta \) based on the quadrant (third quadrant means \( \sin \beta < 0 \)).

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

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주요 개념

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Cosine of a Difference Formula

The cosine of the difference of two angles α and β is given by cos(α - β) = cos α cos β + sin α sin β. This identity allows us to express cos(α - β) in terms of the sines and cosines of α and β individually, which is essential for finding the exact value when given trigonometric values of α and β.

Determining the Sign of Trigonometric Functions Based on Quadrants

The signs of sine and cosine depend on the quadrant in which the angle lies. For example, if π < α < 3π/2 (third quadrant), both sine and cosine are negative. Understanding the quadrant helps determine the correct sign of the unknown trigonometric values, which is crucial for accurate calculation.

Using Pythagorean Identity to Find Missing Values

When either sine or cosine of an angle is given, the other can be found using the Pythagorean identity sin²θ + cos²θ = 1. This is important when only one trigonometric value is provided, enabling the calculation of the other value needed to apply the cosine difference formula.

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β) sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.