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 5.37
Textbook Question
Find one value of θ or x that satisfies each of the following.
tan θ = cot(45° + 2θ)
Verified step by step guidance1
Start by recalling the identity for cotangent: \( \cot(\alpha) = \frac{1}{\tan(\alpha)} \).
Rewrite the equation \( \tan \theta = \cot(45^\circ + 2\theta) \) using the identity: \( \tan \theta = \frac{1}{\tan(45^\circ + 2\theta)} \).
Set up the equation \( \tan \theta \cdot \tan(45^\circ + 2\theta) = 1 \).
Use the tangent addition formula: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \) to express \( \tan(45^\circ + 2\theta) \).
Solve the resulting equation for \( \theta \) by isolating \( \theta \) and considering the periodicity of the tangent function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent and Cotangent Functions
The tangent function, tan(θ), is defined as the ratio of the opposite side to the adjacent side in a right triangle. The cotangent function, cot(θ), is the reciprocal of the tangent function, expressed as cot(θ) = 1/tan(θ). Understanding these functions is crucial for solving equations involving them, as they relate angles to their respective ratios.
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Angle Addition Formulas
The angle addition formulas allow us to express trigonometric functions of sums of angles in terms of the functions of the individual angles. For example, cot(45° + 2θ) can be rewritten using the cotangent addition formula, which helps simplify the equation and find the value of θ. Mastery of these formulas is essential for manipulating and solving trigonometric equations.
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Solving Trigonometric Equations
Solving trigonometric equations involves finding the angle(s) that satisfy a given trigonometric identity. This often requires using algebraic techniques, identities, and sometimes inverse trigonometric functions. In this case, equating tan(θ) to cot(45° + 2θ) and manipulating the equation will lead to the solution for θ, highlighting the importance of systematic problem-solving in trigonometry.
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