Domanda del libro di testo
Verify that each equation is an identity.
(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
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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 5.46
Concept Check Suppose that sec θ = (x+4)/x.
Find an expression in x for tan θ.
Guida verificata passo dopo passo1
Recall the trigonometric identity: \( \sec \theta = \frac{1}{\cos \theta} \).
Given \( \sec \theta = \frac{x+4}{x} \), we can express \( \cos \theta \) as \( \cos \theta = \frac{x}{x+4} \).
Use the Pythagorean identity: \( \tan^2 \theta = \sec^2 \theta - 1 \).
Substitute \( \sec \theta = \frac{x+4}{x} \) into the identity to find \( \tan^2 \theta = \left(\frac{x+4}{x}\right)^2 - 1 \).
Simplify the expression for \( \tan^2 \theta \) and then take the square root to find \( \tan \theta \).
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Concetti chiave
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Secant Function
The secant function, denoted as sec θ, is the reciprocal of the cosine function. It is defined as sec θ = 1/cos θ. In this context, sec θ = (x+4)/x implies a relationship between the angle θ and the variable x, which can be used to derive other trigonometric functions.
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Percorso guidato
6:22
Graphs of Secant and Cosecant Functions
Pythagorean Identity
The Pythagorean identity states that for any angle θ, the relationship sin² θ + cos² θ = 1 holds true. This identity can be rearranged to express tan θ in terms of sec θ, as tan² θ = sec² θ - 1. Understanding this identity is crucial for deriving tan θ from sec θ.
Video consigliato:
Percorso guidato
6:25Pythagorean Identities
Tangent Function
The tangent function, denoted as tan θ, is defined as the ratio of the sine and cosine functions: tan θ = sin θ/cos θ. It can also be expressed in terms of secant as tan θ = √(sec² θ - 1). This relationship allows us to find an expression for tan θ using the given sec θ value.
Video consigliato:
Percorso guidato
5:43Introduction to Tangent Graph
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Domanda del libro di testo
Find sinθ.
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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.
1 + cot(-θ)/cot(-θ)
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Domanda del libro di testo
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Find the remaining five trigonometric functions of θ.
csc θ = -5/2, θ in quadrant III
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Domanda del libro di testo
Match each expression in Column I with its value in Column II.
8. tan (-π/8)
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