Textbook Question
Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]
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 5.1.58
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
Recall the definition of the secant function in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
Rewrite the given expression \(\sin \theta \sec \theta\) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so it becomes \(\sin \theta \times \frac{1}{\cos \theta}\).
Express the product as a single fraction: \(\frac{\sin \theta}{\cos \theta}\).
Recognize that \(\frac{\sin \theta}{\cos \theta}\) is the definition of the tangent function, \(\tan \theta\).
Therefore, the expression simplifies to \(\tan \theta\), which is written only in terms of sine and cosine without any quotients involving secant.
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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 Definitions
Trigonometric functions like sine, cosine, and secant relate angles to ratios of sides in a right triangle or points on the unit circle. Sine (sin θ) is the ratio of the opposite side to the hypotenuse, cosine (cos θ) is adjacent over hypotenuse, and secant (sec θ) is the reciprocal of cosine, i.e., 1/cos θ.
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Introduction to Trigonometric Functions
Expressing Functions in Terms of Sine and Cosine
Many trigonometric expressions can be rewritten using only sine and cosine because these are the fundamental functions. For example, sec θ can be expressed as 1/cos θ. This helps simplify expressions by reducing them to a common set of functions.
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5:53Graph of Sine and Cosine Function
Simplification of Trigonometric Expressions
Simplifying trigonometric expressions involves eliminating quotients and rewriting all terms in a consistent form, often using algebraic manipulation and identities. The goal is to express the function without fractions and only in terms of sine and cosine of θ, making it easier to analyze or solve.
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6:36Simplifying Trig Expressions
Related Practice
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.
(sec²θ - 1)/(csc²θ - 1)
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Textbook Question
Find sinθ.
cot θ = -1/3, θ in quadrant IV
Textbook Question
Verify that each equation is an identity.
sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x
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.
[1 - sin²(-θ)]/[1 + cot²(-θ)]
Textbook Question
Perform each indicated operation and simplify the result so that there are no quotients.
(1 + tan θ)² - 2 tan θ