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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.RE.39
Capitolo 7, Problema 6.RE.39

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
tan x = cot x

Guida verificata passo dopo passo
1
Recall the definitions of tangent and cotangent: \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\).
Set the equation \(\tan x = \cot x\) and rewrite it using the definitions: \(\frac{\sin x}{\cos x} = \frac{\cos x}{\sin x}\).
Cross-multiply to eliminate the fractions: \(\sin^2 x = \cos^2 x\).
Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to express one function in terms of the other, or recognize that \(\sin^2 x = \cos^2 x\) implies \(\sin^2 x - \cos^2 x = 0\).
Rewrite the equation as \(\sin^2 x - \cos^2 x = 0\), which can be factored or recognized as \(\cos 2x = 0\). Solve \(\cos 2x = 0\) over the interval \([0, 2\pi)\) to find the values of \(x\).

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Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal trigonometric functions, where tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x). Understanding their relationship helps in transforming or equating expressions involving these functions.
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Introduction to Cotangent Graph

Solving Trigonometric Equations

Solving equations like tan(x) = cot(x) involves manipulating the equation to find values of x that satisfy it within a given interval, often by using identities or rewriting functions in terms of sine and cosine.
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How to Solve Linear Trigonometric Equations

Interval and General Solutions in Trigonometry

Trigonometric functions are periodic, so solutions repeat every 2π. When solving over [0, 2π), it is important to find all unique solutions within this interval, considering the periodicity and domain restrictions.
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Fundamental Trigonometric Identities
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Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.

sin 2θ = cos 2θ +1

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Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.

sin² θ + 3 sin θ + 2 = 0

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Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.

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Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.

tan² 2x -1 = 0

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Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.

2 tan x -1 = 0

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Solve each equation for all exact solutions, in radians.

cos 2x + cos x = 0

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