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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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