Textbook Question
Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
x + y = 9, 2x + y = -1
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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 88
Verify that each equation is an identity.
sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)
Verified step by step guidance1
Start by recalling the algebraic identity for the sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, let \(a = \sin \theta\) and \(b = \cos \theta\).
Rewrite the left side using the sum of cubes formula: \(\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)\).
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to simplify the expression inside the parentheses: replace \(\sin^2 \theta + \cos^2 \theta\) with 1.
After substitution, the expression becomes \((\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta)\), which matches the right side of the given equation.
Conclude that since both sides simplify to the same expression, the given equation is an identity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often using known formulas like Pythagorean identities or algebraic manipulations.
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Fundamental Trigonometric Identities
Algebraic Factorization
Algebraic factorization involves rewriting expressions as products of simpler factors. Recognizing patterns such as sum of cubes, a³ + b³ = (a + b)(a² - ab + b²), helps in breaking down complex trigonometric expressions to verify identities efficiently.
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Pythagorean Identity
The Pythagorean identity, sin²θ + cos²θ = 1, is fundamental in trigonometry. It allows substitution and simplification of expressions involving sine and cosine powers, which is essential when verifying or transforming trigonometric equations.
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Related Practice
Textbook Question
Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
5x - 2y + 4 = 0, 3x + 5y = 6
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Textbook Question
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
sin 162°
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