Find values of the sine and cosine functions for each angle measure.

θ, given cos 2θ = 2/3 and 90° < θ <180°

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Identify the given information: \( \cos 2\theta = \frac{2}{3} \) and the angle \( \theta \) lies in the second quadrant, i.e., \( 90^\circ < \theta < 180^\circ \).

Recall the double-angle identity for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \). Use this to express \( \cos^2 \theta \) in terms of \( \cos 2\theta \).

Substitute \( \cos 2\theta = \frac{2}{3} \) into the identity and solve for \( \cos^2 \theta \): \[ \frac{2}{3} = 2\cos^2 \theta - 1 \] Rearrange to find \( \cos^2 \theta \).

Determine \( \cos \theta \) by taking the square root of \( \cos^2 \theta \). Since \( \theta \) is in the second quadrant, \( \cos \theta \) is negative.

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \). Since \( \theta \) is in the second quadrant, \( \sin \theta \) is positive.

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Key Concepts

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

Double-Angle Identity for Cosine

The double-angle identity relates cos 2θ to cos θ and sin θ, typically expressed as cos 2θ = cos²θ - sin²θ or cos 2θ = 2cos²θ - 1. This identity allows us to find the values of sine and cosine for θ when cos 2θ is known.

Quadrant and Sign of Trigonometric Functions

Knowing the quadrant of angle θ is crucial because it determines the signs of sine and cosine. Since 90° < θ < 180°, θ lies in the second quadrant where sine is positive and cosine is negative, guiding the correct sign assignment for the values.

Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, connects sine and cosine values. After finding one value using the double-angle formula, this identity helps calculate the other, ensuring the sine and cosine values satisfy this fundamental relationship.

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