Textbook Question
Use the given information to find each of the following.
cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.36
Textbook Question
Simplify each expression.
√[(1 + cos 165°)/(1 - cos 165°)]
Verified step by step guidance1
Recognize that the expression \( \sqrt{\frac{1 + \cos 165^\circ}{1 - \cos 165^\circ}} \) can be simplified using trigonometric identities.
Use the identity \( \cos(180^\circ - \theta) = -\cos(\theta) \) to find \( \cos 165^\circ \).
Substitute \( \cos 165^\circ = -\cos 15^\circ \) into the expression.
The expression becomes \( \sqrt{\frac{1 - \cos 15^\circ}{1 + \cos 15^\circ}} \).
Use the identity \( \frac{1 - \cos \theta}{1 + \cos \theta} = \tan^2\left(\frac{\theta}{2}\right) \) to further simplify the expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. Understanding these identities is crucial for simplifying trigonometric expressions, as they allow for the transformation of functions into more manageable forms.
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Cosine Function Properties
The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. Its values range from -1 to 1, and it exhibits symmetry about the y-axis. Knowing the properties of the cosine function, including its values at specific angles, is essential for evaluating and simplifying expressions involving cosine.
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Rationalizing Expressions
Rationalizing expressions involves eliminating radicals from the denominator of a fraction. This process often simplifies calculations and makes expressions easier to work with. In the context of trigonometric expressions, rationalizing can help in simplifying complex fractions that contain square roots, leading to clearer and more concise results.
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