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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 3.RE.38c
Chapter 3, Problem 3.RE.38c

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

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1
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.
Determine \( \cos \alpha \) using the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Substitute \( \sin \alpha = -\frac{1}{3} \) and solve for \( \cos \alpha \). Since \( \alpha \) is in the third quadrant, \( \cos \alpha \) will be negative.
Determine \( \sin \beta \) using the Pythagorean identity: \( \sin^2 \beta + \cos^2 \beta = 1 \). Substitute \( \cos \beta = -\frac{1}{3} \) and solve for \( \sin \beta \). Since \( \beta \) is in the third quadrant, \( \sin \beta \) will be negative.
Calculate \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} \) using the values found in previous steps.
Use the tangent addition formula to find \( \tan(\alpha + \beta) \): \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \] Substitute the values of \( \tan \alpha \) and \( \tan \beta \) to express \( \tan(\alpha + \beta) \) exactly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 can be derived from their sine and cosine values.
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Sum and Difference of Tangent

Determining Quadrants and Sign of Trigonometric Functions

Knowing the quadrant in which an angle lies is essential to determine the signs of sine, cosine, and tangent. For example, if π < α < 3π/2, α is in the third quadrant where sine and cosine are negative, but tangent is positive. This helps correctly assign signs when calculating tan α and tan β.
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Introduction to Trigonometric Functions

Using Pythagorean Identity to Find Missing Trigonometric Values

Given sin α or cos β, the corresponding cosine α or sine β can be found using the identity sin²θ + cos²θ = 1. This is crucial for computing tan α = sin α / cos α or tan β = sin β / cos β, enabling the use of the sum formula for tangent.
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Pythagorean Identities
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