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 12

Chapter 3, Problem 12

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.

Verified step by step guidance

insert step 1: Understand that \( \cot \theta = 3 \) implies \( \tan \theta = \frac{1}{3} \) because \( \tan \theta = \frac{1}{\cot \theta} \).

insert step 2: Since \( \theta \) is in quadrant III, both sine and cosine are negative, but tangent is positive.

insert step 3: Use the double angle identity for tangent: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).

insert step 4: Substitute \( \tan \theta = \frac{1}{3} \) into the double angle identity: \( \tan 2\theta = \frac{2 \times \frac{1}{3}}{1 - (\frac{1}{3})^2} \).

insert step 5: Simplify the expression to find \( \tan 2\theta \).

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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 that involve trigonometric functions and are true for all values of the variables involved. One important identity is the double angle formula for tangent, which states that tan(2θ) = 2tan(θ) / (1 - tan²(θ)). Understanding these identities is crucial for simplifying and solving trigonometric expressions.

Quadrants and Signs of Trigonometric Functions

The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions. In quadrant III, both sine and cosine are negative, which means tangent (the ratio of sine to cosine) is positive. Knowing the quadrant in which the angle lies helps determine the signs of the trigonometric values needed for calculations.

Finding Trigonometric Values from Cotangent

Cotangent is the reciprocal of tangent, defined as cot(θ) = 1/tan(θ). Given cot(θ) = 3, we can find tan(θ) as 1/3. This relationship allows us to derive other trigonometric values, such as sine and cosine, using the Pythagorean identity, which is essential for calculating tan(2θ) accurately.

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

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Find all solutions of each equation. tan x = 1

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