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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 8a
Chapter 3, Problem 8a

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

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 sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\). Substitute the known values of \(\sin \theta\) and \(\cos \theta\) to express \(\sin 2\theta\).

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Key Concepts

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

Understanding the Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each determining the sign of sine, cosine, and tangent functions. In quadrant II, sine values are positive while cosine and tangent values are negative. Knowing the quadrant helps determine the correct sign of trigonometric values when solving problems.
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Introduction to Trigonometric Functions

Double-Angle Identity for Sine

The double-angle identity for sine states that sin(2θ) = 2 sin(θ) cos(θ). This formula allows you to find the sine of twice an angle using the sine and cosine of the original angle. It is essential for problems requiring exact values of trigonometric functions at multiple angles.
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Double Angle Identities

Using the Pythagorean Identity to Find Cosine

Given sin(θ), the cosine can be found using the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Rearranging gives cos(θ) = ±√(1 - sin²(θ)). The sign depends on the quadrant of θ, which is crucial for determining the correct cosine value to use in calculations.
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Pythagorean Identities
Related Practice
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. Write the expression as the cosine of an angle.

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

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

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

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

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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