Textbook Question
In Exercises 53–62, solve each equation on the interval [0, 2𝝅). (tan x - 1) (cos x + 1) = 0
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Linear Trigonometric Equations
4:39 minutesProblem 3.5.61
Textbook Question
In Exercises 53–62, solve each equation on the interval [0, 2𝝅). tan² x cos x = tan² x
Verified step by step guidance1
Start by writing down the given equation: \(\tan^{2} x \cos x = \tan^{2} x\).
Bring all terms to one side to set the equation equal to zero: \(\tan^{2} x \cos x - \tan^{2} x = 0\).
Factor out the common term \(\tan^{2} x\): \(\tan^{2} x (\cos x - 1) = 0\).
Set each factor equal to zero to find possible solutions: \(\tan^{2} x = 0\) or \(\cos x - 1 = 0\).
Solve each equation separately on the interval \([0, 2\pi)\):
- For \(\tan^{2} x = 0\), find \(x\) such that \(\tan x = 0\).
- For \(\cos x - 1 = 0\), find \(x\) such that \(\cos x = 1\).
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4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Equations
Trigonometric equations involve expressions with trigonometric functions like sine, cosine, and tangent. Solving these equations means finding all angle values within a given interval that satisfy the equation. Understanding how to manipulate and simplify these equations is essential for finding correct solutions.
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Properties of Tangent and Cosine Functions
The tangent function, tan x, is defined as sin x / cos x and has vertical asymptotes where cos x = 0. Cosine, cos x, oscillates between -1 and 1. Recognizing where these functions are zero or undefined helps in solving equations involving tan² x and cos x, especially when factoring or dividing terms.
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Solving Equations on a Specific Interval [0, 2π)
When solving trigonometric equations on the interval [0, 2π), solutions must be found within one full rotation of the unit circle. This requires identifying all angles in this range that satisfy the equation, considering periodicity and the behavior of the trigonometric functions involved.
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