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Ch. 3 - Trigonometric Identities and Equations
Chapter 3, Problem 9

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 24 cos θ = -------- , θ lies in quadrant IV. 25

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<Step 1: Understand the given information.> We are given that \( \cos \theta = \frac{24}{25} \) and \( \theta \) is in quadrant IV. In quadrant IV, cosine is positive, and sine is negative.
<Step 2: Use the Pythagorean identity to find \( \sin \theta \).> The identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = \frac{24}{25} \) into the identity: \( \sin^2 \theta + \left(\frac{24}{25}\right)^2 = 1 \).
<Step 3: Solve for \( \sin \theta \).> Calculate \( \left(\frac{24}{25}\right)^2 \) and subtract it from 1 to find \( \sin^2 \theta \). Then take the square root to find \( \sin \theta \), remembering that \( \sin \theta \) is negative in quadrant IV.
<Step 4: Use the double angle formula for tangent.> The formula for \( \tan 2\theta \) is \( \frac{2 \tan \theta}{1 - \tan^2 \theta} \). First, find \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
<Step 5: Substitute \( \tan \theta \) into the double angle formula.> Use the value of \( \tan \theta \) found in Step 4 to calculate \( \tan 2\theta \) using the formula \( \frac{2 \tan \theta}{1 - \tan^2 \theta} \).

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. In this context, knowing the cosine value allows us to find the sine and tangent values using the Pythagorean identity, which is essential for calculating tan 2θ.
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Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For tangent, the formula is tan 2θ = 2tan θ / (1 - tan² θ). This formula is crucial for finding tan 2θ once we determine tan θ from the given cosine value.
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Quadrants and Angle Signs

Understanding the unit circle and the signs of trigonometric functions in different quadrants is vital. Since θ is in quadrant IV, cosine is positive while sine and tangent are negative. This knowledge helps in accurately determining the values of sine and tangent, which are necessary for calculating tan 2θ.
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