Textbook Question
Find one value of θ or x that satisfies each of the following.
tan θ = cot(45° + 2θ)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.40
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
Recognize that the equation \( \cos x = \sin(\pi/12) \) can be rewritten using the co-function identity: \( \cos x = \cos(\pi/2 - x) \).
Apply the co-function identity to rewrite \( \sin(\pi/12) \) as \( \cos(\pi/2 - \pi/12) \).
Simplify \( \pi/2 - \pi/12 \) to find the angle in terms of cosine: \( \pi/2 - \pi/12 = 5\pi/12 \).
Set the equation \( \cos x = \cos(5\pi/12) \) and solve for \( x \).
Use the property that if \( \cos A = \cos B \), then \( A = B + 2k\pi \) or \( A = -B + 2k\pi \) for any integer \( k \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. The cosine function, cos(x), gives the ratio of the adjacent side to the hypotenuse, while the sine function, sin(x), gives the ratio of the opposite side to the hypotenuse. Understanding these functions is essential for solving equations involving angles.
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Co-function Identity
The co-function identity states that sin(θ) = cos(π/2 - θ) for any angle θ. This identity is crucial when solving equations that involve both sine and cosine, as it allows us to express one function in terms of the other, facilitating the solution process. In this case, it helps relate sin(π/12) to a cosine function.
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Unit Circle
The unit circle is a fundamental concept in trigonometry that defines the sine and cosine of angles based on their coordinates on a circle with a radius of one. Each point on the unit circle corresponds to an angle, where the x-coordinate represents the cosine and the y-coordinate represents the sine. This geometric representation aids in visualizing and solving trigonometric equations.
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