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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 9c
Chapter 3, Problem 9c

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

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 \(\cos \theta = \frac{24}{25}\) to find \(\sin \theta\).
Calculate \(\sin \theta\) by rearranging the identity: \(\sin \theta = -\sqrt{1 - \left(\frac{24}{25}\right)^2}\), taking the negative root because \(\sin \theta\) is negative in quadrant IV.
Use the double-angle formula for tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\). To apply this, first find \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Substitute \(\tan \theta\) into the double-angle formula to express \(\tan 2\theta\) in terms of known values, then simplify the expression to find the exact value.

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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 side lengths. Knowing the quadrant of the angle is crucial because it determines the sign (positive or negative) of these ratios. In quadrant IV, cosine is positive while sine and tangent are negative.
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Quadratic Formula

Double-Angle Identity for Tangent

The double-angle identity for tangent states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This formula allows you to find the tangent of twice an angle using the tangent of the original angle, which is essential for solving the problem once tan θ is known.
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Double Angle Identities

Finding tan θ from cos θ

Given cos θ, you can find sin θ using the Pythagorean identity sin² θ + cos² θ = 1. Then, tan θ is calculated as sin θ divided by cos θ. The quadrant information helps determine the correct sign of sin θ and tan θ.
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Finding Components from Direction and Magnitude
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: 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

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

8
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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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