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
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.6
Textbook Question
For each expression in Column I, choose the expression from Column II that completes an identity.
6. sec² x = ____
Verified step by step guidance1
Recall the Pythagorean identity: \( \sec^2 x = 1 + \tan^2 x \).
Identify that \( \sec^2 x \) is related to \( \tan^2 x \) through this identity.
Understand that \( \sec^2 x \) is the reciprocal of \( \cos^2 x \), which is part of the identity.
Recognize that the identity \( \sec^2 x = 1 + \tan^2 x \) is derived from the fundamental Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \).
Choose the expression \( 1 + \tan^2 x \) from Column II to complete the identity for \( \sec^2 x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Secant Function
The secant function, denoted as sec(x), is the reciprocal of the cosine function. It is defined as sec(x) = 1/cos(x). Understanding secant is crucial for solving trigonometric identities, as it relates directly to the cosine function and can be expressed in terms of sine and cosine.
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Pythagorean Identity
Pythagorean identities are fundamental relationships in trigonometry that involve the squares of sine, cosine, and tangent functions. The most common identity is sin²(x) + cos²(x) = 1. This identity can be manipulated to express sec²(x) in terms of tangent, specifically sec²(x) = 1 + tan²(x), which is essential for completing the given expression.
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Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved. They are used to simplify expressions and solve equations in trigonometry. Recognizing and applying these identities, such as the relationship between secant and tangent, is key to completing the expression in the question accurately.
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