Textbook Question
Simplify each expression.
±√[(1 + cos 20α)/2]
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.44
Textbook Question
Simplify each expression.
±√[(1 - cos (3θ/5))/2]
Verified step by step guidance1
Recognize that the expression involves a trigonometric identity related to the cosine function.
Recall the half-angle identity for cosine: \( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \).
Notice that the given expression \( \pm \sqrt{\frac{1 - \cos \left( \frac{3\theta}{5} \right)}{2}} \) resembles the half-angle identity for sine: \( \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} \).
Conclude that the expression simplifies to \( \sin \left( \frac{3\theta}{10} \right) \) using the half-angle identity for sine.
Understand that the \( \pm \) sign indicates the principal value, which depends on the specific quadrant of \( \frac{3\theta}{10} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine Function
The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic and oscillates between -1 and 1. Understanding the properties of the cosine function is essential for simplifying expressions involving angles, especially in the context of trigonometric identities.
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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 identity, angle sum and difference identities, and double angle formulas. These identities are crucial for simplifying trigonometric expressions and solving equations, as they allow for the transformation of one form into another.
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Square Root and Simplification
The square root operation is the inverse of squaring a number, and it is often used in trigonometric simplifications. In the expression ±√[(1 - cos(3θ/5))/2], recognizing that this represents the sine of half the angle (via the half-angle identity) is key. Simplifying expressions involving square roots requires careful manipulation and understanding of how to apply identities effectively.
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