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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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.

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05:06

Double Angle Identities

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.

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Pythagorean Identities

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.

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