Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.36

Chapter 6, Problem 5.36

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot t tan t

Verified step by step guidance

Recognize that \( \cot t \) is the cotangent of \( t \), which is defined as \( \frac{1}{\tan t} \).

Substitute \( \cot t \) with \( \frac{1}{\tan t} \) in the expression \( \cot t \tan t \).

The expression becomes \( \frac{1}{\tan t} \times \tan t \).

Apply the property of multiplication that states \( a \times \frac{1}{a} = 1 \) for any non-zero \( a \).

Conclude that \( \frac{1}{\tan t} \times \tan t = 1 \), simplifying the expression to a constant.

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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 that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, serve as the foundation for simplifying trigonometric expressions. Understanding these identities is crucial for manipulating and simplifying expressions like 'cot t tan t'.

Cotangent and Tangent Functions

The cotangent (cot) and tangent (tan) functions are fundamental trigonometric functions defined as the ratios of the sides of a right triangle. Specifically, cotangent is the reciprocal of tangent, expressed as cot t = 1/tan t. Recognizing the relationship between these functions is essential for simplifying expressions involving them, such as 'cot t tan t', which simplifies to 1.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic manipulation to reduce complex expressions to simpler forms. This process often includes factoring, combining like terms, and substituting equivalent expressions. In the case of 'cot t tan t', applying the identity that cot t is the reciprocal of tan t leads to a straightforward simplification to 1.

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Simplifying Trig Expressions

Related Practice

Textbook Question

Verify that each equation is an identity.

sin² θ (1 + cot² θ) - 1 = 0

Textbook Question

Factor each trigonometric expression.

sec² θ - 1

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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.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

Textbook Question

Match each expression in Column I with its value in Column II.

10. cos 67.5°

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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

Perform each transformation. See Example 2.

Write cot x in terms of sin x.

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