Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 3.5.61

Chapter 3, Problem 3.5.61

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). tan² x cos x = tan² x

Verified step by step guidance

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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Key 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.

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.

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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Related Practice

Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). sin x + 2 sin x cos x = 0

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Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 cos² x + 3 cos x + 1 = 0

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In Exercises 59–68, verify each identity.

cos²(θ/2) = (sec θ + 1)/(2 sec θ)

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Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). (2 cos x + √ 3) (2 sin x + 1) = 0

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In Exercises 47–54, use the figures to find the exact value of each trigonometric function. 2sin(θ/2)cos(θ/2)

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Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan(x/2) = √3

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