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

검증된 단계별 안내

Rewrite the given equation in terms of \( \sin \theta \): \[ 6 \sin^{2} \theta + \sin \theta = 1 \].

Bring all terms to one side to form a quadratic equation in \( \sin \theta \): \[ 6 \sin^{2} \theta + \sin \theta - 1 = 0 \].

Let \( u = \sin \theta \). Substitute to get \( 6u^{2} + u - 1 = 0 \).

Solve the quadratic equation \( 6u^{2} + u - 1 = 0 \) using the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a=6 \), \( b=1 \), and \( c=-1 \).

For each solution \( u \), check if it lies within the valid range for sine (\( -1 \leq u \leq 1 \)). Then solve for \( \theta \) by finding \( \theta = \arcsin(u) \) and use the unit circle to find all solutions within the specified domain, expressing answers in degrees and radians as required.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Solving Quadratic Trigonometric Equations

This involves rewriting the trigonometric equation in a quadratic form, such as a sin²θ + b sinθ + c = 0, and then solving for the trigonometric function (e.g., sinθ) using algebraic methods like factoring or the quadratic formula.

General Solutions for Sine Equations

Once sinθ is found, the general solutions for θ are determined using the unit circle properties: θ = arcsin(value) + 2πn and θ = π - arcsin(value) + 2πn for integer n, ensuring all possible angles are considered.

Converting Between Radians and Degrees and Using Least Nonnegative Angles

Understanding how to convert between radians and degrees is essential for expressing answers correctly. Additionally, solutions should be expressed as the smallest nonnegative angle within one full rotation (0 to 2π radians or 0° to 360°), rounding as specified.