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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 9b
Chapter 3, Problem 9b

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

Verified step by step guidance
1
Identify the given information: \(\cos \theta = \frac{24}{25}\) and \(\theta\) lies in quadrant IV.
Recall the double-angle identity for cosine: \(\cos 2\theta = 2\cos^{2} \theta - 1\) or equivalently \(\cos 2\theta = 1 - 2\sin^{2} \theta\).
Since \(\cos \theta\) is given, calculate \(\sin \theta\) using the Pythagorean identity \(\sin^{2} \theta + \cos^{2} \theta = 1\). Substitute \(\cos \theta = \frac{24}{25}\) to find \(\sin \theta\).
Determine the sign of \(\sin \theta\) in quadrant IV. Remember that in quadrant IV, sine is negative, so take the negative root for \(\sin \theta\).
Use the double-angle formula \(\cos 2\theta = 2\cos^{2} \theta - 1\) and substitute the known value of \(\cos \theta\) to find \(\cos 2\theta\).

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Key Concepts

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

Cosine of an Angle in a Specific Quadrant

The cosine function measures the horizontal coordinate on the unit circle. Knowing the quadrant of the angle helps determine the sign of cosine; in quadrant IV, cosine values are positive, which confirms the given positive value of cos θ = 24/25.
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Quadratic Formula

Double-Angle Identity for Cosine

The double-angle formula for cosine states that cos 2θ = 2 cos² θ - 1 or cos 2θ = 1 - 2 sin² θ. This identity allows us to find the exact value of cos 2θ using the known value of cos θ or sin θ.
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Double Angle Identities

Using Pythagorean Identity to Find sin θ

The Pythagorean identity sin² θ + cos² θ = 1 helps find sin θ when cos θ is known. Since cos θ = 24/25, sin θ can be calculated as ±√(1 - (24/25)²). The sign of sin θ depends on the quadrant; in quadrant IV, sin θ is negative.
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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\(\theta\)

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

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

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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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