Domanda del libro di testo
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
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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 5.66
Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0
Guida verificata passo dopo passo1
Start by recalling the Pythagorean identity: \(1 + \cot^2 \theta = \csc^2 \theta\).
Substitute \(1 + \cot^2 \theta\) with \(\csc^2 \theta\) in the equation: \(\sin^2 \theta \cdot \csc^2 \theta - 1 = 0\).
Remember that \(\csc \theta = \frac{1}{\sin \theta}\), so \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\).
Substitute \(\csc^2 \theta\) with \(\frac{1}{\sin^2 \theta}\) in the equation: \(\sin^2 \theta \cdot \frac{1}{\sin^2 \theta} - 1 = 0\).
Simplify the expression: \(1 - 1 = 0\), which confirms the identity.
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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 verifying equations, as they allow us to manipulate and simplify expressions to show equivalence.
Video consigliato:
Percorso guidato
5:32
Fundamental Trigonometric Identities
Cotangent Function
The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). It can also be expressed in terms of sine and cosine, which is essential for transforming and simplifying trigonometric expressions. Recognizing how cotangent relates to sine and cosine is vital for verifying identities involving cotangent.
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Percorso guidato
5:37Introduction to Cotangent Graph
Pythagorean Identity
The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This fundamental identity is often used to simplify trigonometric expressions and verify identities. In the context of the given equation, recognizing how to apply this identity can help in transforming the left-hand side to demonstrate that it equals zero.
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Percorso guidato
6:25Pythagorean Identities
Pratica correlata
Domanda del libro di testo
For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
-tan x cos x
II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x
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Domanda del libro di testo
Find the remaining five trigonometric functions of θ.
cos θ = 1/5, θ in quadrant I
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Domanda del libro di testo
Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
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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.
cos θ (cos θ - sec θ)
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Domanda del libro di testo
Perform each transformation. See Example 2.
Write cot x in terms of sin x.
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