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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 9
Chapter 3, Problem 9

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

Verified step by step guidance
1
Identify the given information: \(\cos \theta = \frac{24}{25}\) and \(\theta\) lies in quadrant IV.
Recall that in quadrant IV, cosine is positive and sine is negative. Use the Pythagorean identity to find \(\sin \theta\): \(\sin^2 \theta + \cos^2 \theta = 1\).
Substitute the known value of \(\cos \theta\) into the identity: \(\sin^2 \theta + \left(\frac{24}{25}\right)^2 = 1\).
Solve for \(\sin \theta\): \(\sin^2 \theta = 1 - \left(\frac{24}{25}\right)^2\), then take the square root and choose the negative root because \(\theta\) is in quadrant IV.
Use the double-angle formula for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) to express \(\sin 2\theta\) 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, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. The quadrant in which the angle lies determines the sign (positive or negative) of these ratios. For example, in quadrant IV, cosine is positive while sine is negative.
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Quadratic Formula

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This identity allows you to find the sine value when cosine is known, by rearranging to sin θ = ±√(1 - cos²θ). The sign depends on the quadrant of θ.
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Pythagorean Identities

Double-Angle Formula for Sine

The double-angle formula for sine is sin 2θ = 2 sin θ cos θ. It expresses the sine of twice an angle in terms of the sine and cosine of the original angle. This formula is essential for finding sin 2θ once sin θ and cos θ are known.
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Related Practice
Textbook Question

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

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

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: 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

Use the given information to find the exact value of each of the following: cos2θ\(\cos\)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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