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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 14c
Chapter 3, Problem 14c

Use the given information to find the exact value of each of the following: tan 2θ
sin θ = ﹣2/3, θ lies in quadrant III.

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Identify the given information: \(\sin \theta = -\frac{2}{3}\) and \(\theta\) lies in quadrant III. Recall that in quadrant III, both sine and cosine are negative, and tangent is positive.
Use the Pythagorean identity to find \(\cos \theta\): \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\sin \theta = -\frac{2}{3}\) to get \(\left(-\frac{2}{3}\right)^2 + \cos^2 \theta = 1\).
Simplify the equation to find \(\cos^2 \theta\): \(\frac{4}{9} + \cos^2 \theta = 1\), so \(\cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9}\). Since \(\theta\) is in quadrant III, \(\cos \theta\) is negative, so \(\cos \theta = -\frac{\sqrt{5}}{3}\).
Calculate \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values found: \(\tan \theta = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}}\).
Use the double-angle formula for tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\). Substitute the expression for \(\tan \theta\) from the previous step and simplify to find the exact value of \(\tan 2\theta\).

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Key Concepts

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

Trigonometric Ratios and Their Signs in Quadrants

Trigonometric functions like sine, cosine, and tangent have specific signs depending on the quadrant of the angle. In quadrant III, both sine and cosine are negative, but tangent, being sine divided by cosine, is positive. Understanding these sign rules helps determine the correct values of trigonometric functions.
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Quadratic Formula

Double-Angle Identity for Tangent

The double-angle formula for tangent is tan(2θ) = (2 tan θ) / (1 - tan² θ). This identity allows us to find the tangent of twice an angle using the tangent of the original angle. It is essential for solving problems involving tan 2θ when only information about θ is given.
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Double Angle Identities

Using Pythagorean Identity to Find Cosine

Given sin θ, the Pythagorean identity sin² θ + cos² θ = 1 helps find cos θ. Since sin θ is negative in quadrant III, cos θ will also be negative. Calculating cos θ accurately is crucial to determine tan θ = sin θ / cos θ, which is needed for the double-angle formula.
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Pythagorean Identities
Related Practice
Textbook Question

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

sin θ = ﹣2/3, θ lies in quadrant III.

Textbook Question

Find all solutions of each equation. cos x = ﹣1/2

Textbook Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

Textbook Question

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

sin θ = ﹣2/3, θ lies in quadrant III.

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 x + sin 2x

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Textbook Question

Use a sum or difference formula to find the exact value of each expression. cos(45° + 30°)