Textbook Question
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.1.72
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.
cos θ (cos θ - sec θ)
Verified step by step guidance1
Start by rewriting the given expression \(\cos \theta (\cos \theta - \sec \theta)\), focusing on expressing all trigonometric functions in terms of sine and cosine.
Recall that \(\sec \theta\) is the reciprocal of \(\cos \theta\), so rewrite \(\sec \theta\) as \(\frac{1}{\cos \theta}\).
Substitute \(\sec \theta\) in the expression to get \(\cos \theta \left( \cos \theta - \frac{1}{\cos \theta} \right)\).
Distribute \(\cos \theta\) across the terms inside the parentheses: \(\cos \theta \cdot \cos \theta - \cos \theta \cdot \frac{1}{\cos \theta}\).
Simplify each term: \(\cos^2 \theta - 1\), ensuring no quotients remain and all functions are in terms of \(\theta\) only.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions and Their Reciprocal Identities
Understanding the basic trigonometric functions sine and cosine, along with their reciprocal functions such as secant (sec θ = 1/cos θ), is essential. This allows rewriting expressions involving secant in terms of cosine, facilitating simplification.
Recommended video:
5:32
Fundamental Trigonometric Identities
Expression Simplification Without Quotients
Simplifying trigonometric expressions to eliminate quotients involves rewriting all terms so that no fractions remain. This often requires multiplying through by denominators or substituting reciprocal identities to express everything in sine and cosine only.
Recommended video:
03:40Quotients of Complex Numbers in Polar Form
Algebraic Manipulation of Trigonometric Expressions
Skillful algebraic manipulation, such as distributing terms, combining like terms, and factoring, is necessary to simplify trigonometric expressions. This helps in rewriting the expression in a cleaner form involving only sine and cosine functions of θ.
Recommended video:
6:36Simplifying Trig Expressions
Related Practice
Textbook Question
Find the remaining five trigonometric functions of θ.
cos θ = 1/5, θ in quadrant I
Textbook Question
Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
Textbook Question
Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2
Textbook Question
Find sinθ.
csc θ = -8/5
Textbook Question
Verify that each equation is an identity.
1/(sec α - tan α) = sec α + tan α
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