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.

√2 sin 3x - 1 = 0

Verified step by step guidance

Start by isolating the sine term in the equation: \( \sqrt{2} \sin 3x - 1 = 0 \). Add 1 to both sides to get \( \sqrt{2} \sin 3x = 1 \).

Next, divide both sides by \( \sqrt{2} \) to solve for \( \sin 3x \): \( \sin 3x = \frac{1}{\sqrt{2}} \).

Recall that \( \sin \theta = \frac{1}{\sqrt{2}} \) corresponds to angles where \( \theta = \frac{\pi}{4} + 2k\pi \) or \( \theta = \frac{3\pi}{4} + 2k\pi \) for any integer \( k \). Here, \( \theta = 3x \).

Set up the two equations for \( 3x \): \( 3x = \frac{\pi}{4} + 2k\pi \) and \( 3x = \frac{3\pi}{4} + 2k\pi \), where \( k \) is any integer.

Solve each equation for \( x \) by dividing both sides by 3: \( x = \frac{\pi}{12} + \frac{2k\pi}{3} \) and \( x = \frac{3\pi}{12} + \frac{2k\pi}{3} \). Then, find all solutions within the desired interval, and convert to degrees if needed, rounding as specified.

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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 and finding all angle values that satisfy the equation within a given domain. Solutions often include general forms using periodicity, and exact or approximate values depending on the problem's requirements.

Angle Measures and Unit Conversion

Understanding the difference between radians and degrees is essential, as well as converting between them. Radians are based on the radius of a circle, while degrees divide a circle into 360 parts. Correct unit usage ensures accurate solutions and proper rounding.

Using Inverse Trigonometric Functions and Periodicity

Inverse sine (arcsin) helps find principal angle solutions, but because sine is periodic with period 2π (or 360°), all solutions must account for this repetition. Writing solutions with the least nonnegative angle measures requires adjusting for these periodic intervals.

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Related Practice

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.

2 cos² x + cos x ― 1 = 0

Textbook Question

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 2π). Express solutions to four decimal places.

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Textbook Question

cos θ + 1 = 0

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Solve each equation for x.

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sin² θ ― 2 sin θ + 3 = 0

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

2 cos 2x = √3