if sinx = 4/5 and x is in quadrant 2, then tan2x = ______. a. 24/7 b. -12/7 c. 12/7 d. -24/7

Answer:

D. $-\frac{24}{7}$

Explanation:

Step1: Find $\cos x$

Given $\sin x=\frac{4}{5}$ and $x$ in quadrant 2. Using $\sin^{2}x+\cos^{2}x = 1$, we have $\cos x=-\sqrt{1 - \sin^{2}x}=-\sqrt{1 - (\frac{4}{5})^{2}}=-\frac{3}{5}$.

Step2: Calculate $\tan x$

By the formula $\tan x=\frac{\sin x}{\cos x}$, so $\tan x=\frac{\frac{4}{5}}{-\frac{3}{5}}=-\frac{4}{3}$.

Step3: Use double - angle formula for tangent

The double - angle formula for tangent is $\tan2x=\frac{2\tan x}{1-\tan^{2}x}$. Substitute $\tan x = -\frac{4}{3}$ into it: $\tan2x=\frac{2\times(-\frac{4}{3})}{1-(-\frac{4}{3})^{2}}=\frac{-\frac{8}{3}}{1-\frac{16}{9}}=\frac{-\frac{8}{3}}{-\frac{7}{9}}=-\frac{24}{7}$.