Textbook Question
Find values of the sine and cosine functions for each angle measure.
2θ, given cos θ = (√3)/5 and sin θ > 0
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6. Trigonometric Identities and More Equations
Double Angle Identities
Problem 5.16
Textbook Question
Find values of the sine and cosine functions for each angle measure.
θ, given cos 2θ = 2/3 and 90° < θ <180°
Verified step by step guidance1
Recognize that the given range for \( \theta \) is \( 90^\circ < \theta < 180^\circ \), which places \( \theta \) in the second quadrant where sine is positive and cosine is negative.
Use the double angle identity for cosine: \( \cos 2\theta = 2\cos^2\theta - 1 \). Substitute \( \cos 2\theta = \frac{2}{3} \) into the equation to get \( \frac{2}{3} = 2\cos^2\theta - 1 \).
Solve the equation \( \frac{2}{3} = 2\cos^2\theta - 1 \) for \( \cos^2\theta \). Rearrange to find \( \cos^2\theta = \frac{5}{6} \).
Since \( \theta \) is in the second quadrant, \( \cos\theta \) is negative. Therefore, \( \cos\theta = -\sqrt{\frac{5}{6}} \).
Use the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) to find \( \sin\theta \). Substitute \( \cos^2\theta = \frac{5}{6} \) into the identity to solve for \( \sin^2\theta \), and then find \( \sin\theta \).
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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) = 2cos²(θ) - 1. This identity is essential for solving problems involving angles that are multiples of a given angle, allowing us to relate the cosine of a double angle to the cosine of the angle itself.
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Quadrants and Sign of Trigonometric Functions
The unit circle is divided into four quadrants, each affecting the sign of the sine and cosine functions. In the second quadrant (90° < θ < 180°), sine is positive while cosine is negative. Understanding the signs of these functions in different quadrants is crucial for determining the correct values of sine and cosine based on the given angle.
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Pythagorean Identity
The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This relationship allows us to find the sine value if we know the cosine value, or vice versa. In this problem, once we determine cos(θ) from cos(2θ), we can use this identity to find sin(θ), ensuring that both values are consistent with the angle's quadrant.
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