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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 23
Chapter 3, Problem 23

Find all solutions of each equation. 3 sin θ + 5 = ﹣2 sin θ

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Start by writing down the given equation: \(3 \sin \theta + 5 = -2 \sin \theta\).
To isolate the sine term, add \(2 \sin \theta\) to both sides to combine like terms: \(3 \sin \theta + 2 \sin \theta + 5 = 0\).
Simplify the left side by combining the sine terms: \(5 \sin \theta + 5 = 0\).
Next, subtract 5 from both sides to isolate the sine term: \(5 \sin \theta = -5\).
Divide both sides by 5 to solve for \(\sin \theta\): \(\sin \theta = -1\).

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Key Concepts

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

Solving Trigonometric Equations

This involves isolating the trigonometric function (like sin θ) and finding all angle values that satisfy the equation within a given domain. Solutions often require algebraic manipulation followed by using inverse trigonometric functions.
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How to Solve Linear Trigonometric Equations

Properties of the Sine Function

The sine function is periodic with period 2π and ranges between -1 and 1. Understanding its behavior helps determine all possible solutions for sin θ, including those in different quadrants where sine has the same value.
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Graph of Sine and Cosine Function

Using Inverse Sine and General Solutions

After isolating sin θ, the inverse sine function (arcsin) gives a principal solution. To find all solutions, one must use the general solution formulas θ = arcsin(x) + 2nπ and θ = π - arcsin(x) + 2nπ, where n is any integer.
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Inverse Sine
Related Practice
Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

Textbook Question

In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t

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Textbook Question

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sin 2x = √3 / 2

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Textbook Question

Find all solutions of each equation. 4 sin θ﹣1 = 2 sin θ

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Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 𝝅/3 + 𝝅/4 )

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 5𝝅/3 ﹣ 𝝅/4)