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.
5 + 5 tan² θ = 6 sec θ

검증된 단계별 안내

Start by rewriting the given equation: \(5 + 5 \tan^{2} \theta = 6 \sec \theta\).

Recall the Pythagorean identity relating tangent and secant: \(1 + \tan^{2} \theta = \sec^{2} \theta\). Use this to express \(\tan^{2} \theta\) in terms of \(\sec^{2} \theta\) as \(\tan^{2} \theta = \sec^{2} \theta - 1\).

Substitute \(\tan^{2} \theta = \sec^{2} \theta - 1\) into the original equation to get \(5 + 5(\sec^{2} \theta - 1) = 6 \sec \theta\).

Simplify the equation to form a quadratic in \(\sec \theta\): \(5 + 5 \sec^{2} \theta - 5 = 6 \sec \theta\), which reduces to \(5 \sec^{2} \theta = 6 \sec \theta\).

Rewrite the equation as \(5 \sec^{2} \theta - 6 \sec \theta = 0\), factor it, and solve for \(\sec \theta\). Then, find the corresponding values of \(\theta\) by considering the definition \(\sec \theta = \frac{1}{\cos \theta}\) and solving for \(\theta\) in the specified domain, ensuring to express solutions in the least possible nonnegative angle measures.

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주요 개념

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

Trigonometric Identities

Trigonometric identities like tan²θ + 1 = sec²θ relate different trig functions and simplify equations. Recognizing and applying these identities helps transform the given equation into a solvable form by expressing all terms in a common function.

Solving Trigonometric Equations

Solving trig equations involves isolating the trig function, finding general solutions using unit circle values, and applying periodicity to list all solutions. Understanding how to handle multiple solutions and restrictions on the domain is essential.

Angle Measurement and Conversion

Angles can be measured in radians or degrees, and converting between them is often necessary. Solutions must be expressed in the least nonnegative angle measure, requiring knowledge of angle normalization and rounding conventions for approximate answers.