Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.30

Chapter 6, Problem 5.30

Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2

Verified step by step guidance

Recognize that the expression \( \cot^4 x + 3 \cot^2 x + 2 \) can be treated as a quadratic in terms of \( \cot^2 x \).

Let \( u = \cot^2 x \). Then the expression becomes \( u^2 + 3u + 2 \).

Factor the quadratic expression \( u^2 + 3u + 2 \) by finding two numbers that multiply to 2 and add to 3.

The numbers 1 and 2 satisfy these conditions, so the expression factors as \( (u + 1)(u + 2) \).

Substitute back \( u = \cot^2 x \) to get the factored form \( (\cot^2 x + 1)(\cot^2 x + 2) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. Understanding these identities, such as the Pythagorean identities, can help simplify and manipulate trigonometric expressions. In this case, recognizing that cotangent can be expressed in terms of sine and cosine may aid in factoring the expression.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. In the given expression, recognizing it as a quadratic in terms of cot² x allows us to apply factoring techniques similar to those used in algebra.

Quadratic Form

A quadratic form is an expression that can be written in the standard form ax² + bx + c. In the context of the given expression, cot⁴ x + 3 cot² x + 2 can be treated as a quadratic in cot² x. This perspective enables the use of factoring methods to find the roots or simplify the expression effectively.

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Related Practice

Textbook Question

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

Textbook Question

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

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Textbook Question

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)

Textbook Question

Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Textbook Question

Verify that each equation is an identity.

1/(sec α - tan α) = sec α + tan α

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Textbook Question

Simplify each expression.

± √[(1 + cos (x/4))/2]