Use the given information to find the exact value of each of the following: sin 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 \(\cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9}\).

Determine the sign of \(\cos \theta\) in quadrant III. Since cosine is negative in quadrant III, \(\cos \theta = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3}\).

Use the double-angle formula for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\). Substitute the values found: \(\sin 2\theta = 2 \times \left(-\frac{2}{3}\right) \times \left(-\frac{\sqrt{5}}{3}\right)\).

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Double-Angle Identity for Sine

The double-angle identity for sine states that sin(2θ) = 2 sin(θ) cos(θ). This formula allows you to find the sine of twice an angle using the sine and cosine of the original angle, which is essential for solving problems involving sin 2θ.

Determining the Sign of Trigonometric Functions by Quadrant

The sign of sine, cosine, and tangent depends on the quadrant in which the angle lies. Since θ is in quadrant III, both sine and cosine are negative. This information helps determine the correct signs of trigonometric values when calculating sin 2θ.

Using the Pythagorean Identity to Find Cosine

Given sin θ, the Pythagorean identity sin²θ + cos²θ = 1 allows you to find cos θ by rearranging to cos θ = ±√(1 - sin²θ). The sign of cos θ is chosen based on the quadrant of θ, which is crucial for accurately computing sin 2θ.

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