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

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

검증된 단계별 안내

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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주요 개념

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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.

Find values of the sine and cosine functions for each angle measure.θ, given cos 2θ = 2/3 and 90° < θ <180°