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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 11a
Chapter 3, Problem 11a

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

Verified step by step guidance
1
Recall the double-angle identity for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\).
Use the given information \(\cot \theta = 2\) to express \(\tan \theta\) as \(\frac{1}{2}\), since \(\cot \theta = \frac{1}{\tan \theta}\).
Determine the signs of \(\sin \theta\) and \(\cos \theta\) in quadrant III, where both sine and cosine are negative.
Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and the value \(\tan \theta = \frac{1}{2}\) to set up a ratio between \(\sin \theta\) and \(\cos \theta\).
Find exact values for \(\sin \theta\) and \(\cos \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), then substitute these into the double-angle formula \(\sin 2\theta = 2 \sin \theta \cos \theta\).

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Key Concepts

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

Cotangent and Its Relationship to Sine and Cosine

Cotangent (cot θ) is the reciprocal of tangent, defined as cot θ = cos θ / sin θ. Knowing cot θ allows us to express sine and cosine in terms of each other, which is essential for finding sin 2θ. Understanding this relationship helps in determining the values of sine and cosine from cotangent.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and tangent functions depends on the quadrant where the angle lies. Since θ is in quadrant III, both sine and cosine are negative, but tangent (and cotangent) is positive. This information is crucial to assign correct signs to sine and cosine values when calculating sin 2θ.
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Quadratic Formula

Double-Angle Identity for Sine

The double-angle identity for sine states that sin 2θ = 2 sin θ cos θ. This formula allows us to find sin 2θ once sin θ and cos θ are known or expressed in terms of cot θ. Applying this identity is the key step in solving the problem.
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Double Angle Identities
Related Practice
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

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Textbook Question

Find all solutions of each equation. sin x = (√3)/2

Textbook Question

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

cot θ = 2, θ lies in quadrant III.

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Textbook Question

Verify each identity. csc θ - sin θ = cot θ cos θ

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Textbook Question
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.
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.