Find one value of θ or x that satisfies each of the following.
tan θ = cot(45° + 2θ)

검증된 단계별 안내

Recall the definition of cotangent in terms of tangent: \(\cot \alpha = \frac{1}{\tan \alpha}\). So the equation \(\tan \theta = \cot(45^\circ + 2\theta)\) can be rewritten as \(\tan \theta = \frac{1}{\tan(45^\circ + 2\theta)}\).

Multiply both sides of the equation by \(\tan(45^\circ + 2\theta)\) to get rid of the fraction: \(\tan \theta \cdot \tan(45^\circ + 2\theta) = 1\).

Use the tangent addition formula to express \(\tan(45^\circ + 2\theta)\): \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Here, \(a = 45^\circ\) and \(b = 2\theta\), so \(\tan(45^\circ + 2\theta) = \frac{\tan 45^\circ + \tan 2\theta}{1 - \tan 45^\circ \tan 2\theta}\).

Substitute \(\tan 45^\circ = 1\) into the expression to simplify: \(\tan(45^\circ + 2\theta) = \frac{1 + \tan 2\theta}{1 - \tan 2\theta}\).

Replace \(\tan(45^\circ + 2\theta)\) in the equation \(\tan \theta \cdot \tan(45^\circ + 2\theta) = 1\) with the simplified expression and solve for \(\theta\): \(\tan \theta \cdot \frac{1 + \tan 2\theta}{1 - \tan 2\theta} = 1\). From here, you can proceed to isolate \(\theta\).

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주요 개념

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

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal trigonometric functions, where cot(α) = 1/tan(α). Understanding this relationship allows us to rewrite cotangent expressions in terms of tangent, facilitating equation solving.

Angle Sum Identities

Angle sum identities express trigonometric functions of sums of angles, such as tan(A + B) = (tan A + tan B) / (1 - tan A tan B). These identities help simplify or transform expressions involving sums like 45° + 2θ.

Solving Trigonometric Equations

Solving trigonometric equations involves manipulating expressions using identities and algebraic techniques to isolate the variable. Recognizing equivalent angles and periodicity is key to finding valid solutions.

Find one value of θ or x that satisfies each of the following.tan θ = cot(45° + 2θ)