Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.46

Chapter 6, Problem 5.46

Concept Check Suppose that sec θ = (x+4)/x.
Find an expression in x for tan θ.

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Recall the trigonometric identity: \( \sec \theta = \frac{1}{\cos \theta} \).

Given \( \sec \theta = \frac{x+4}{x} \), we can express \( \cos \theta \) as \( \cos \theta = \frac{x}{x+4} \).

Use the Pythagorean identity: \( \tan^2 \theta = \sec^2 \theta - 1 \).

Substitute \( \sec \theta = \frac{x+4}{x} \) into the identity to find \( \tan^2 \theta = \left(\frac{x+4}{x}\right)^2 - 1 \).

Simplify the expression for \( \tan^2 \theta \) and then take the square root to find \( \tan \theta \).

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Key Concepts

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

Secant Function

The secant function, denoted as sec θ, is the reciprocal of the cosine function. It is defined as sec θ = 1/cos θ. In this context, sec θ = (x+4)/x implies a relationship between the angle θ and the variable x, which can be used to derive other trigonometric functions.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the relationship sin² θ + cos² θ = 1 holds true. This identity can be rearranged to express tan θ in terms of sec θ, as tan² θ = sec² θ - 1. Understanding this identity is crucial for deriving tan θ from sec θ.

Tangent Function

The tangent function, denoted as tan θ, is defined as the ratio of the sine and cosine functions: tan θ = sin θ/cos θ. It can also be expressed in terms of secant as tan θ = √(sec² θ - 1). This relationship allows us to find an expression for tan θ using the given sec θ value.

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Related Practice

Textbook Question

Determine whether the positive or negative square root should be selected.

cos 58° = ±√ (1 + cos 116°)/2]

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.

1 + cot(-θ)/cot(-θ)

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.

csc θ - sin θ

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Textbook Question

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

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Textbook Question

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1

Textbook Question

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

8. tan (-π/8)