Find one value of θ or x that satisfies each of the following.
cos x = sin (π/12)

검증된 단계별 안내

Recall the co-function identity in trigonometry: \(\cos x = \sin \left( \frac{\pi}{2} - x \right)\).

Use the identity to rewrite the equation \(\cos x = \sin \left( \frac{\pi}{12} \right)\) as \(\cos x = \cos \left( \frac{\pi}{2} - \frac{\pi}{12} \right)\).

Simplify the angle inside the cosine on the right side: \(\frac{\pi}{2} - \frac{\pi}{12} = \frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12}\), so the equation becomes \(\cos x = \cos \left( \frac{5\pi}{12} \right)\).

Recall that if \(\cos A = \cos B\), then \(A = B + 2k\pi\) or \(A = -B + 2k\pi\) for any integer \(k\).

Set up the two equations: \(x = \frac{5\pi}{12} + 2k\pi\) and \(x = -\frac{5\pi}{12} + 2k\pi\), then choose an integer \(k\) (usually \(k=0\)) to find one specific value of \(x\).

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주요 개념

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

Relationship Between Sine and Cosine

Sine and cosine functions are co-functions, meaning sin(θ) = cos(π/2 - θ). This identity allows us to rewrite one trigonometric function in terms of the other, which is useful for solving equations like cos x = sin(π/12).

Unit Circle and Angle Measures

The unit circle represents angles and their sine and cosine values. Understanding how angles correspond to points on the circle helps in finding all possible solutions for trigonometric equations within a given interval.

Solving Basic Trigonometric Equations

Solving equations like cos x = sin(π/12) involves using identities and inverse functions to find angle values. Recognizing multiple solutions due to periodicity is important when determining all valid values of x.

Find one value of θ or x that satisfies each of the following.cos x = sin (π/12)