Domanda del libro di testo
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.
csc θ cos θ tan θ
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 5.76
Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
Guida verificata passo dopo passo1
Step 1: Start by simplifying the left-hand side of the equation. Combine the fractions \( \frac{1 + \sin \theta}{1 - \sin \theta} - \frac{1 - \sin \theta}{1 + \sin \theta} \) by finding a common denominator, which is \((1 - \sin \theta)(1 + \sin \theta)\).
Step 2: Rewrite each fraction with the common denominator: \( \frac{(1 + \sin \theta)^2 - (1 - \sin \theta)^2}{(1 - \sin \theta)(1 + \sin \theta)} \).
Step 3: Expand the numerators: \((1 + \sin \theta)^2 = 1 + 2\sin \theta + \sin^2 \theta\) and \((1 - \sin \theta)^2 = 1 - 2\sin \theta + \sin^2 \theta\).
Step 4: Subtract the expanded forms: \(1 + 2\sin \theta + \sin^2 \theta - (1 - 2\sin \theta + \sin^2 \theta)\), which simplifies to \(4\sin \theta\).
Step 5: The denominator \((1 - \sin \theta)(1 + \sin \theta)\) simplifies to \(1 - \sin^2 \theta = \cos^2 \theta\). Thus, the left-hand side becomes \(\frac{4\sin \theta}{\cos^2 \theta}\), which simplifies to \(4 \tan \theta \sec \theta\), verifying the identity.
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Concetti chiave
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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. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations as identities.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Simplifying Expressions
Simplifying trigonometric expressions involves manipulating the equation using algebraic techniques and trigonometric identities to make it easier to analyze or prove. This may include combining fractions, factoring, or using identities to rewrite terms. Mastery of simplification techniques is essential for verifying the validity of trigonometric equations.
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Percorso guidato
6:36Simplifying Trig Expressions
Tangent and Secant Functions
The tangent (tan) and secant (sec) functions are fundamental trigonometric functions defined as tan θ = sin θ / cos θ and sec θ = 1 / cos θ, respectively. These functions are often used in identities and equations involving angles. Understanding their relationships and how to manipulate them is key to solving and verifying trigonometric identities.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
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Verify that each equation is an identity.
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Domanda del libro di testo
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Verify that each equation is an identity.
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