Question

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

Accepted Answer

Identify the given information: $$\sin 	heta = -\frac{2}{3}$$ and $$	heta$$ 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 $$\cos 	heta$$: $$\sin^2$ 	heta + \cos^2$ 	heta = 1. $$Substitute $$\sin 	heta = -\frac{2}{3} to $$get $$\left(-\frac{2}{3}\$$right)^2$$ + \cos^2$ 	heta = 1. $$Simplify the equation to find $$\cos^2$ 	heta$$: $$\frac{4}{9} + \cos^2$ 	heta = 1$$, so $$\cos^2$ 	heta = 1 - \frac{4}{9} = \frac{5}{9}. $$Since $$	heta is in $$quadrant III, $$\cos 	heta is $$negative, so $$\cos 	heta = -\frac{\sqrt{5}}{3}. $$Calculate $$	an 	heta$$ using the definition $$	an 	heta = \frac{\sin 	heta}{\cos 	heta}. $$Substitute the values found: $$	an 	heta = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}}. $$Use the double-angle formula for tangent: $$	an 2	heta = \frac{2 	an 	heta}{1 - 	an^2$ 	heta}. $$Substitute the expression for $$	an 	heta$$ from the previous step and simplify to find the exact value of $$	an 2	heta.$$

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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Key Concepts

Trigonometric Ratios and Their Signs in Quadrants

Double-Angle Identity for Tangent

Using Pythagorean Identity to Find Cosine