Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θcot θ = 2, θ lies in quadrant III.
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6. Trigonometric Identities and More Equations
Double Angle Identities
2:43 minutesProblem 14
Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ2sin θ = ﹣ -------- , θ lies in quadrant III.3
Verified step by step guidance1
insert step 1: Recognize that since \( \sin \theta = -\frac{2}{3} \) and \( \theta \) is in quadrant III, both sine and cosine are negative in this quadrant.
insert step 2: Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).
insert step 3: Substitute \( \sin \theta = -\frac{2}{3} \) into the identity: \( \left(-\frac{2}{3}\right)^2 + \cos^2 \theta = 1 \).
insert step 4: Solve for \( \cos^2 \theta \) and then find \( \cos \theta \), remembering that \( \cos \theta \) is negative in quadrant III.
insert step 5: Use the double angle formula for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \) 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.
Double Angle Formulas
Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For cosine, the formula is cos(2θ) = cos²(θ) - sin²(θ) or alternatively, cos(2θ) = 2cos²(θ) - 1. Understanding these formulas is essential for calculating values like cos(2θ) when given sin(θ) or cos(θ).
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Trigonometric Values in Quadrants
The unit circle defines the signs of trigonometric functions in different quadrants. In quadrant III, both sine and cosine values are negative. This is crucial for determining the correct signs of sin(θ) and cos(θ) when calculating cos(2θ) based on the given information about θ's location.
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Pythagorean Identity
The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This identity allows us to find the cosine value when we know the sine value, especially useful in this problem where sin(θ) is provided. By rearranging the identity, we can derive cos(θ) and subsequently use it to find cos(2θ).
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