Use the given information to find the exact value of each of the following: tan 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, and tangent is positive.

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 to find \(\cos^2 \theta\): \(\frac{4}{9} + \cos^2 \theta = 1\), so \(\cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9}\). Since \(\theta\) is in quadrant III, \(\cos \theta\) is negative, so \(\cos \theta = -\frac{\sqrt{5}}{3}\).

Calculate \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values found: \(\tan \theta = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}}\).

Use the double-angle formula for tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\). Substitute the expression for \(\tan \theta\) from the previous step and simplify to find the exact value of \(\tan 2\theta\).

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Trigonometric Ratios and Their Signs in Quadrants

Trigonometric functions like sine, cosine, and tangent have specific signs depending on the quadrant of the angle. In quadrant III, both sine and cosine are negative, but tangent, being sine divided by cosine, is positive. Understanding these sign rules helps determine the correct values of trigonometric functions.

Double-Angle Identity for Tangent

The double-angle formula for tangent is tan(2θ) = (2 tan θ) / (1 - tan² θ). This identity allows us to find the tangent of twice an angle using the tangent of the original angle. It is essential for solving problems involving tan 2θ when only information about θ is given.

Using Pythagorean Identity to Find Cosine

Given sin θ, the Pythagorean identity sin² θ + cos² θ = 1 helps find cos θ. Since sin θ is negative in quadrant III, cos θ will also be negative. Calculating cos θ accurately is crucial to determine tan θ = sin θ / cos θ, which is needed for the double-angle formula.

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