Textbook Question
In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x = √ 3 sin x
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.RE.35a
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.
Verified step by step guidance1
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 quadrants for \( \alpha \) and \( \beta \) 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:
\[ \cos \alpha = +\sqrt{1 - \left(\frac{3}{5}\right)^2} \]
\[ \cos \beta = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \]
Apply the sine addition formula:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
Substitute the known values of \( \sin \alpha \), \( \cos \alpha \), \( \sin \beta \), and \( \cos \beta \) into the formula to express \( \sin(\alpha + \beta) \) 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 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 sine and cosine of each individual angle, which is essential for solving the problem.
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Verifying Identities with Sum and Difference Formulas
Determining Cosine from Sine and Quadrant
Given sin θ and the quadrant of angle θ, we can find cos θ using the Pythagorean identity cos² θ = 1 - sin² θ. The sign of cos θ depends on the quadrant, so knowing the angle's range is crucial to assign the correct positive or negative value.
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Understanding Angle Ranges and Quadrants
The given angle ranges (e.g., 0 < α < π/2) indicate the quadrant in which each angle lies. This information helps determine the signs of sine and cosine values, which is necessary for correctly applying trigonometric identities and finding exact values.
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6:36Quadratic Formula
Related Practice
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In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos 2x = -1
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