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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 8b
Chapter 3, Problem 8b

Use the given information to find the exact value of each of the following: cos2θ\(\cos\)2\(\theta\)
sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

Verified step by step guidance
1
Identify the given information: \(\sin \theta = \frac{12}{13}\) and \(\theta\) lies in quadrant II.
Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Use this to find \(\cos \theta\).
Calculate \(\cos \theta\) by rearranging the identity: \(\cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - \left(\frac{12}{13}\right)^2}\).
Determine the correct sign of \(\cos \theta\) based on the quadrant. Since \(\theta\) is in quadrant II, \(\cos \theta\) is negative.
Use the double-angle formula for cosine: \(\cos 2\theta = 2 \cos^2 \theta - 1\). Substitute the value of \(\cos \theta\) found in the previous step to express \(\cos 2\theta\).

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Key Concepts

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

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. Given sin θ, this identity allows you to find cos θ by rearranging the equation to cos²θ = 1 - sin²θ. This is essential for determining the cosine value when only sine is known.
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Pythagorean Identities

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle lies. In quadrant II, sine is positive and cosine is negative. This knowledge helps assign the correct sign to cos θ after calculating its magnitude using the Pythagorean identity.
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Quadratic Formula

Double-Angle Formula for Cosine

The double-angle formula for cosine is cos 2θ = 2 cos²θ - 1 or cos 2θ = 1 - 2 sin²θ. This formula allows you to find cos 2θ using either sine or cosine of θ, making it crucial for solving the problem when sin θ is given.
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Double Angle Identities
Related Practice
Textbook Question

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\)

sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

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Textbook Question

Use the given information to find the exact value of each of the following: tan2θ\(\tan\)2\(\theta\)

sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

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Textbook Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

cos5π12cosπ12+sin5π12sinπ12\(\cos\) \(\frac{5\pi}{12}\) \(\cos\) \(\frac{\pi}{12}\) + \(\sin\) \(\frac{5\pi}{12}\) \(\sin\) \(\frac{\pi}{12}\)

Textbook Question

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

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Textbook Question

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\)

cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)

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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 6x + sin 2x

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