Textbook Question
Verify that each equation is an identity.
(2 tan B)/(sin 2B) = sec² B
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.58
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
sin θ sec θ
Verified step by step guidance1
Start by expressing sec \( \theta \) in terms of sine and cosine. Recall that \( \sec \theta = \frac{1}{\cos \theta} \).
Substitute \( \sec \theta \) in the expression \( \sin \theta \sec \theta \) with \( \frac{1}{\cos \theta} \).
The expression becomes \( \sin \theta \cdot \frac{1}{\cos \theta} \).
Simplify the expression by multiplying: \( \frac{\sin \theta}{\cos \theta} \).
Recognize that \( \frac{\sin \theta}{\cos \theta} \) is the definition of \( \tan \theta \). Therefore, the expression simplifies to \( \tan \theta \).
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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 (sin) and cosine (cos), are fundamental in trigonometry. They relate the angles of a triangle to the ratios of its sides. Understanding these functions is crucial for manipulating and simplifying expressions involving angles, particularly in the context of right triangles.
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Secant Function
The secant function (sec) is defined as the reciprocal of the cosine function, expressed as sec θ = 1/cos θ. This relationship is essential for rewriting expressions that involve secant in terms of sine and cosine. Recognizing this reciprocal relationship allows for simplification of trigonometric expressions.
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Simplification of Trigonometric Expressions
Simplifying trigonometric expressions involves rewriting them to eliminate quotients and express them solely in terms of sine and cosine. This process often includes using identities and reciprocal relationships, which helps in making the expressions easier to work with and understand, especially in solving equations or evaluating limits.
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