Textbook Question
Use identities to fill in each blank with the appropriate trigonometric function name.
____ 72° = cot 18°
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 40
Textbook Question
Find one value of θ or x that satisfies each of the following.
cos x = sin (π/12)
Verified step by step guidance1
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).
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Amplitude and Reflection of Sine and Cosine
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.
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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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