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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.35b
Chapter 3, Problem 3.RE.35b

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Verified step by step guidance
1
Identify the given information: \( \sin \alpha = \frac{3}{5} \) with \( 0 < \alpha < \frac{\pi}{2} \), and \( \sin \beta = \frac{12}{13} \) with \( \frac{\pi}{2} < \beta < \pi \).
Determine the quadrant of each angle to find the signs of \( \cos \alpha \) and \( \cos \beta \). Since \( \alpha \) is in the first quadrant, \( \cos \alpha > 0 \). Since \( \beta \) is in the second quadrant, \( \cos \beta < 0 \).
Use the Pythagorean identity to find \( \cos \alpha \) and \( \cos \beta \): \[ \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \] Calculate: \[ \cos \alpha = +\sqrt{1 - \left(\frac{3}{5}\right)^2} \] \[ \cos \beta = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \]
Recall the cosine difference identity: \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]
Substitute the values of \( \sin \alpha \), \( \sin \beta \), \( \cos \alpha \), and \( \cos \beta \) into the identity to express \( \cos(\alpha - \beta) \) exactly.

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Key Concepts

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

Sum and Difference Formulas for Cosine

The cosine of the difference of two angles, cos(α - β), can be found using the formula cos(α - β) = cos α cos β + sin α sin β. This identity allows us to express the cosine of a difference in terms of the sines and cosines of the individual angles, which is essential when given sine values and angle ranges.
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Verifying Identities with Sum and Difference Formulas

Determining Cosine from Sine and Quadrant Information

Given sin α and sin β along with their angle ranges, we can find cos α and cos β using the Pythagorean identity cos²θ = 1 - sin²θ. The sign of cosine depends on the quadrant of the angle, so knowing the interval for α and β helps determine whether cosine is positive or negative.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Angle Measurement in Radians and Quadrant Boundaries

Angles are given in radians with specified intervals (e.g., 0 < α < 3π/2). Understanding these intervals helps identify the quadrant in which each angle lies, which is crucial for determining the signs of trigonometric functions and correctly applying identities.
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Converting between Degrees & Radians
Related Practice
Textbook Question

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). sin² θ - 1 = 0

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Textbook Question

In Exercises 43–44, express each product as a sum or difference. sin 7x cos 3x

Textbook Question

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x - sin x - 1 = 0

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Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions:

a. sin(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Textbook Question

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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Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions:

d. sin 2α

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

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