In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

검증된 단계별 안내

Identify the given information: $ \sin \alpha = \frac{5\pi}{6} $ with $ \frac{3\pi}{2} < \alpha < \pi $, and $ \tan \beta = \frac{7}{2} $ with $ \pi < \beta < \frac{3\pi}{2} $. Note that the interval for $ \alpha $ seems inconsistent since $ \frac{3\pi}{2} > \pi $. Verify the correct interval for $ \alpha $ before proceeding.

Recall the formula for the tangent of a sum: $\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

Find $ \tan \alpha $ using the given $ \sin \alpha $ and the Pythagorean identity. Since $ \sin \alpha = \frac{5\pi}{6} $, use: $\cos \alpha = \pm \sqrt{1 - \sin^2 \alpha}$ Determine the sign of $ \cos \alpha $ based on the quadrant of $ \alpha $. Then calculate $ \(\tan$ $\alpha$ = $\frac{\sin \alpha}{\cos \alpha}$$.

Use the given $ \tan \beta = \frac{7}{2} $ directly, noting the sign of $ \tan \beta $ based on the quadrant of $ \beta $.

Substitute $ \tan \alpha $ and $ \tan \beta $ into the tangent sum formula: $\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$ Simplify the expression to find the exact value.

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주요 개념

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Sum of Angles Formula for Tangent

The tangent of the sum of two angles α and β is given by tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows us to find the exact value of tan(α + β) using the individual tangents of α and β, which is essential for solving the problem.

Determining the Sign and Value of Trigonometric Functions in Specific Quadrants

Knowing the quadrant in which an angle lies helps determine the sign (positive or negative) of sine, cosine, and tangent values. Since α and β are restricted to certain intervals, this information is crucial to correctly evaluate the trigonometric functions and avoid sign errors.

Finding Tangent from Sine and Quadrant Information

Given sin α and the quadrant of α, we can find cos α using the Pythagorean identity cos²α = 1 - sin²α, then compute tan α = sin α / cos α. This step is necessary when only sine is provided, enabling the use of the sum formula for tangent.