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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Identify the given information: \( \cos \theta = \frac{\sqrt{3}}{5} \) and \( \sin \theta > 0 \). Since \( \sin \theta > 0 \), \( \theta \) is in the first quadrant where sine is positive.

Use the Pythagorean identity to find \( \sin \theta \): \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = \frac{\sqrt{3}}{5} \) into the identity to get \( \sin^2 \theta = 1 - \left( \frac{\sqrt{3}}{5} \right)^2 \).

Calculate \( \sin \theta \) by taking the positive square root (since \( \sin \theta > 0 \)): \( \sin \theta = \sqrt{1 - \frac{3}{25}} \).

Use the double-angle formulas to find \( \sin 2\theta \) and \( \cos 2\theta \): \( \sin 2\theta = 2 \sin \theta \cos \theta \) and \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).

Substitute the values of \( \sin \theta \) and \( \cos \theta \) into the double-angle formulas to express \( \sin 2\theta \) and \( \cos 2\theta \) in terms of known quantities.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2θ in terms of functions of θ. For sine, sin(2θ) = 2 sin θ cos θ, and for cosine, cos(2θ) = cos² θ - sin² θ. These formulas allow calculation of sine and cosine values for double angles using known values of θ.

Pythagorean Identity

The Pythagorean identity states that sin² θ + cos² θ = 1 for any angle θ. Given cos θ, this identity helps find sin θ by rearranging to sin θ = ±√(1 - cos² θ). The sign depends on the quadrant where θ lies, which is crucial for determining the correct sine value.

Quadrant and Sign of Trigonometric Functions

The sign of sine and cosine depends on the quadrant of the angle θ. Since sin θ > 0 and cos θ = √3/5 (positive), θ lies in the first quadrant where both sine and cosine are positive. This information ensures correct sign selection when calculating sine and cosine values.

Find values of the sine and cosine functions for each angle measure.2θ, given cos θ = (√3)/5 and sin θ > 0