question 19 of 20 if sin x = 4/5 and x is in quadrant 2, then tan2x = ______. a. 12/7 b. 24/7 c. -24/7 d…

Answer:

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

Explanation:

Step1: Find $\cos x$

Since $\sin^{2}x+\cos^{2}x = 1$ and $\sin x=\frac{4}{5}$, then $\cos^{2}x=1 - \sin^{2}x=1-\left(\frac{4}{5}\right)^{2}=1-\frac{16}{25}=\frac{9}{25}$. As $x$ is in quadrant 2, $\cos x=-\frac{3}{5}$.

Step2: Find $\tan x$

$\tan x=\frac{\sin x}{\cos x}=\frac{\frac{4}{5}}{-\frac{3}{5}}=-\frac{4}{3}$.

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 the formula: $\tan2x=\frac{2\times\left(-\frac{4}{3}\right)}{1-\left(-\frac{4}{3}\right)^{2}}=\frac{-\frac{8}{3}}{1-\frac{16}{9}}=\frac{-\frac{8}{3}}{-\frac{7}{9}}=-\frac{24}{7}$.