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 11c

Chapter 3, Problem 11c

Use the given information to find the exact value of each of the following: tan 2θ
cot θ = 2, θ lies in quadrant III.

Verified step by step guidance

Recall the double-angle identity for tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).

Substitute \(\tan 2\theta\) into the expression \(\tan 2\theta \cot \theta\) to get \(\left( \frac{2 \tan \theta}{1 - \tan^2 \theta} \right) \cdot \cot \theta\).

Rewrite \(\cot \theta\) as \(\frac{1}{\tan \theta}\) and simplify the expression: \(\left( \frac{2 \tan \theta}{1 - \tan^2 \theta} \right) \cdot \frac{1}{\tan \theta} = \frac{2}{1 - \tan^2 \theta}\).

Set the simplified expression equal to 2 (given): \(\frac{2}{1 - \tan^2 \theta} = 2\) and solve for \(\tan^2 \theta\).

Use the fact that \(\theta\) lies in quadrant III, where tangent is positive, to determine the exact value of \(\tan \theta\) from \(\tan^2 \theta\).

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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 hold true for all values within their domains. For example, the double-angle identity for tangent, tan(2θ) = 2 tan θ / (1 - tan² θ), is essential for expressing tan 2θ in terms of tan θ, enabling simplification and solving of equations.

Quadrant Sign Rules

The sign of trigonometric functions depends on the quadrant in which the angle lies. In quadrant III, both sine and cosine are negative, making tangent positive. Understanding these sign conventions is crucial for determining the correct values of trigonometric functions and their ratios.

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal functions: cot θ = 1 / tan θ. This relationship allows conversion between the two, simplifying expressions like tan 2θ cot θ. Recognizing this reciprocal nature helps in manipulating and solving trigonometric equations efficiently.

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Related Practice

Textbook Question

Solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + cos x = 0

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Textbook Question

Use the given information to find the exact value of each of the following: sin 2θ

cot θ = 2, θ lies in quadrant III.

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Textbook Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin 7x ﹣ sin 3x

In Exercises 7–14, use the given information to find the exact value of each of the following:c. tan 2θcot θ = 3, θ lies in quadrant III.

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Textbook Question

Use the given information to find the exact value of each of the following: cos 2θ

cot θ = 2, θ lies in quadrant III.