Textbook Question
Find the remaining five trigonometric functions of θ.
csc θ = -5/2, θ in quadrant III
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 5.52
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
Verified step by step guidance1
Start by recalling the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \).
Recall the identity for tangent and secant: \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) and \( \sec^2 \alpha = 1 + \tan^2 \alpha \).
Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the equation: \( \sin^2 \alpha + \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha = \sec^2 \alpha \).
Combine the terms on the left side over a common denominator: \( \frac{\sin^2 \alpha \cdot \cos^2 \alpha + \sin^2 \alpha + \cos^4 \alpha}{\cos^2 \alpha} \).
Simplify the expression and verify if it equals \( \sec^2 \alpha \), using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Pythagorean Identity
The Pythagorean identity states that for any angle α, the relationship sin² α + cos² α = 1 holds true. This fundamental identity is crucial in trigonometry as it connects the sine and cosine functions, allowing for simplifications and transformations in various equations.
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Pythagorean Identities
Definition of Tangent and Secant
Tangent and secant are defined in terms of sine and cosine: tan α = sin α / cos α and sec α = 1 / cos α. Understanding these definitions is essential for manipulating trigonometric identities, as they allow us to express equations in terms of sine and cosine, facilitating verification of identities.
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6:22Graphs of Secant and Cosecant Functions
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Recognizing and applying these identities is key to verifying equations and simplifying trigonometric expressions.
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Related Practice
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1 - 1/sec² x
Textbook Question
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
Textbook Question
Find sinθ.
sec θ = 7/2, tan θ < 0
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.
tan θ cos θ
Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
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