Use the given information to find the exact value of each of the following: cos 2θ
sin θ = ﹣2/3, θ lies in quadrant III.

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Identify the given information: \(\sin \theta = -\frac{2}{3}\) and \(\theta\) lies in quadrant III. Recall that in quadrant III, both sine and cosine are negative.

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

Simplify the equation: \(\frac{4}{9} + \cos^2 \theta = 1\), then solve for \(\cos^2 \theta\) to find \(\cos^2 \theta = 1 - \frac{4}{9}\).

Calculate \(\cos \theta\) by taking the square root of \(\cos^2 \theta\). Since \(\theta\) is in quadrant III, \(\cos \theta\) is negative, so choose the negative root.

Use the double-angle formula for cosine: \(\cos 2\theta = 2 \cos^2 \theta - 1\). Substitute the value of \(\cos^2 \theta\) found earlier to express \(\cos 2\theta\) exactly.

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Trigonometric Ratios and Quadrants

Trigonometric ratios like sine and cosine relate angles to side lengths in right triangles. The sign of these ratios depends on the quadrant where the angle lies. Since θ is in quadrant III, both sine and cosine values are negative, which affects the calculation of cos 2θ.

Double-Angle Identity for Cosine

The double-angle identity for cosine states that cos 2θ = 1 - 2sin²θ or cos 2θ = 2cos²θ - 1. This formula allows finding the cosine of twice an angle using the sine or cosine of the original angle, which is essential when only sin θ is given.

Using Pythagorean Identity to Find cos θ

The Pythagorean identity sin²θ + cos²θ = 1 helps find cos θ when sin θ is known. Since sin θ is given, cos θ can be calculated as ±√(1 - sin²θ), with the sign determined by the quadrant of θ. This step is crucial for applying the double-angle formula correctly.

Use the given information to find the exact value of each of the following: cos 2θ sin θ = ﹣2/3, θ lies in quadrant III.