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√3 sin x/2 = 3

검증된 단계별 안내

Start by isolating the sine function in the equation: given \( 2\sqrt{3} \sin \frac{x}{2} = 3 \), divide both sides by \( 2\sqrt{3} \) to get \( \sin \frac{x}{2} = \frac{3}{2\sqrt{3}} \).

Simplify the right-hand side by rationalizing the denominator if needed, to express \( \sin \frac{x}{2} \) in its simplest exact form.

Next, find the general solutions for \( \frac{x}{2} \) by using the inverse sine function: \( \frac{x}{2} = \sin^{-1}(\text{value}) \). Remember that sine is positive in the first and second quadrants, so consider both \( \theta = \sin^{-1}(\text{value}) \) and \( \theta = \pi - \sin^{-1}(\text{value}) \) for solutions in radians.

Write the general solutions for \( x \) by multiplying both sides of the equation by 2: \( x = 2\theta + 2k\pi \), where \( k \) is any integer, to account for the periodicity of sine.

Finally, express the solutions using the least possible nonnegative angle measures by finding the principal values within one full rotation (0 to \( 2\pi \)) and convert these radian solutions to degrees if required, rounding approximate answers as specified.

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주요 개념

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Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. This often requires using inverse trigonometric functions and considering the periodic nature of sine, cosine, or tangent to find all solutions.

Unit Circle and Angle Measures

The unit circle relates angles to coordinates on a circle of radius one, helping to determine sine and cosine values. Understanding angle measures in radians and degrees, and converting between them, is essential for interpreting solutions and expressing answers in the required units.

General Solutions for Sine Equations

Sine equations have infinitely many solutions due to periodicity. The general solution for sin θ = k is θ = arcsin(k) + 2nπ or θ = π - arcsin(k) + 2nπ, where n is any integer. Recognizing this helps find all exact solutions and express them using the least nonnegative angles.

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√3 sin x/2 = 3