In Exercises 35–38, find the exact value of the following under the given conditions:
a. sin(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: \( \sin \alpha = -\frac{1}{3} \) with \( \pi < \alpha < \frac{3\pi}{2} \), and \( \cos \beta = -\frac{1}{3} \) with \( \pi < \beta < \frac{3\pi}{2} \). Both angles are in the third quadrant.

Recall the sine addition formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \). We need to find \( \cos \alpha \) and \( \sin \beta \) to use this formula.

Use the Pythagorean identity to find \( \cos \alpha \): \( \cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \). The negative sign is because \( \alpha \) is in the third quadrant where cosine is negative.

Similarly, find \( \sin \beta \) using the identity \( \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \), since \( \beta \) is also in the third quadrant where sine is negative.

Substitute all known values into the sine addition formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \), and simplify the expression to find the exact value.

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주요 개념

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Sum of Angles Formula for Sine

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

Determining Cosine and Sine from Given Values and Quadrants

Given sin α and cos β along with their quadrant information, we use the Pythagorean identity (sin²θ + cos²θ = 1) to find the missing cosine or sine values. The quadrant determines the sign (positive or negative) of these values, which is crucial for accuracy.

Understanding Angle Measures and Quadrants

The problem specifies angle ranges (π < α < 3π/2 and π < β < 3π/2), indicating both angles lie in the third quadrant. In this quadrant, sine and cosine values are negative, which affects the sign of the trigonometric functions and must be considered when calculating exact values.

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