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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 14b
Chapter 3, Problem 14b

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.

Verified step by step guidance
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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.
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: \(\frac{4}{9} + \cos^2 \theta = 1\), then solve for \(\cos^2 \theta\) to find \(\cos^2 \theta = 1 - \frac{4}{9}\).
Calculate \(\cos \theta\) by taking the square root of \(\cos^2 \theta\). Since \(\theta\) is in quadrant III, \(\cos \theta\) is negative, so choose the negative root.
Use the double-angle formula for cosine: \(\cos 2\theta = 2 \cos^2 \theta - 1\). Substitute the value of \(\cos^2 \theta\) found earlier to express \(\cos 2\theta\) exactly.

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Key Concepts

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

Trigonometric Ratios and Quadrants

Trigonometric ratios like sine and cosine relate angles to side lengths in right triangles. The sign of these ratios depends on the quadrant where the angle lies. Since θ is in quadrant III, both sine and cosine values are negative, which affects the calculation of cos 2θ.
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Double-Angle Identity for Cosine

The double-angle identity for cosine states that cos 2θ = 1 - 2sin²θ or cos 2θ = 2cos²θ - 1. This formula allows finding the cosine of twice an angle using the sine or cosine of the original angle, which is essential when only sin θ is given.
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Using Pythagorean Identity to Find cos θ

The Pythagorean identity sin²θ + cos²θ = 1 helps find cos θ when sin θ is known. Since sin θ is given, cos θ can be calculated as ±√(1 - sin²θ), with the sign determined by the quadrant of θ. This step is crucial for applying the double-angle formula correctly.
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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.

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

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Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

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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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