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.
sin θ sec 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 5.86
Verify that each equation is an identity.
sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x
Guida verificata passo dopo passo1
Start by recalling the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\).
Express \(\cot x\) and \(\tan x\) in terms of \(\sin x\) and \(\cos x\): \(\cot x = \frac{\cos x}{\sin x}\) and \(\tan x = \frac{\sin x}{\cos x}\).
Substitute \(\cot x\) and \(\tan x\) into the equation: \(\sin^2 x(1 + \frac{\cos x}{\sin x}) + \cos^2 x(1 - \frac{\sin x}{\cos x}) + \cot^2 x\).
Simplify each term: \(\sin^2 x + \sin x \cos x + \cos^2 x - \sin x \cos x + \cot^2 x\).
Use the identity \(\cot^2 x = \csc^2 x - 1\) to simplify further and verify 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 and simplifying expressions in trigonometry.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Reciprocal Functions
Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). For example, csc x = 1/sin x and cot x = 1/tan x. Recognizing these relationships helps in transforming and simplifying trigonometric expressions.
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Percorso guidato
3:23Secant, Cosecant, & Cotangent on the Unit Circle
Simplification Techniques
Simplification techniques involve manipulating trigonometric expressions using identities to make them easier to analyze or verify. This can include factoring, combining like terms, or substituting equivalent expressions. Mastery of these techniques is essential for proving that an equation is an identity.
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Percorso guidato
6:20Example 6
Pratica correlata
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.
(sec²θ - 1)/(csc²θ - 1)
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Domanda del libro di testo
Use a half-angle identity to find each exact value.
sin 195°
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Domanda del libro di testo
Use identities to correctly complete each sentence.
If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.
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Domanda del libro di testo
Find sinθ.
cot θ = -1/3, θ in quadrant IV
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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 - sin²(-θ)]/[1 + cot²(-θ)]
841
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