Question

Find sin(2theta) if tan(theta) = - 12/5 and 3pi/2 < theta < 2pi

Answer 1

\[\tan \theta=\frac{-12}{5}=\frac{\sin \theta}{\sqrt{1-\sin^2 \theta}}\]\[\sin^2 \theta=\frac{(-12)^2}{5^2}(1-\sin^2 \theta)\]\[(1+\frac{144}{25})\sin^2 \theta=\frac{144}{25}\]\[\sin \theta=\sqrt{\frac{\frac{144}{25}}{1+\frac{144}{25}}}\]
This will simplify easily.
Also, sine is negative in the fourth quadrant. So your answer is the negative of the value above.
In fact, the answer is -12/13.

Answer 2

SORRYYYY. I found out sinθ. You can now easily find cosθ.
And the answer is 2sinθcosθ = sin(2θ)