Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problem 6.2.57

Chapter 7, Problem 6.2.57

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
sin x (3 sin x - 1) = 1

Verified step by step guidance

Start by rewriting the given equation: \(\sin x (3 \sin x - 1) = 1\).

Expand the left side to get a quadratic form in terms of \(\sin x\): \(3 \sin^2 x - \sin x = 1\).

Rearrange the equation to standard quadratic form: \(3 \sin^2 x - \sin x - 1 = 0\).

Use the quadratic formula to solve for \(\sin x\), where \(a = 3\), \(b = -1\), and \(c = -1\). The formula is \(\sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Find all values of \(x\) (in radians and degrees) that satisfy the solutions for \(\sin x\), considering the domain of the sine function and using the least possible nonnegative angle measures. Round approximate answers as instructed.

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Key Concepts

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

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values that satisfy the given equation. This often requires algebraic manipulation, factoring, and using inverse trigonometric functions. Solutions must consider the periodic nature of trig functions, leading to multiple valid answers within specified intervals.

Radians and Degrees Conversion

Radians and degrees are two units for measuring angles. Understanding how to convert between them (1 radian = 180/π degrees) is essential for interpreting and expressing solutions correctly. Problems may require answers in both units, rounded appropriately as specified.

Using the Unit Circle and Reference Angles

The unit circle helps identify exact values of sine, cosine, and tangent for common angles. Reference angles allow finding all solutions within a cycle by relating angles to their acute counterparts. This is crucial for expressing solutions as least nonnegative angles and for determining all possible solutions.

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Reference Angles on the Unit Circle

Related Practice

Textbook Question

Answer each question.

Suppose solving a trigonometric equation for solutions over the interval [0, 2π) leads to 2x = 2π/3, 2π, 8π/3. What are the corresponding values of x?

Textbook Question

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

6 sin² θ + sin θ = 1

Textbook Question

Solve for exact solutions over the interval [0°, 360°).

sin θ/2 = -√3/2

Textbook Question

cos θ/2 = 1

Textbook Question

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

9 sin² θ ― 6 sin² θ = 1

Textbook Question

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

sin x = sin 2x