# If tanx=-3/4 and 3Π/2<x<2Π,then value of sin2x?

Apr 2, 2018

$\sin \left(2 x\right) = - \frac{24}{25}$

#### Explanation:

We seek $\sin \left(2 x\right)$ when $\tan \left(x\right) = - \frac{3}{4}$

By the double angle identity

color(blue)(sin(2x)=2sin(x)cos(x)

Let's express this in term of tangens

$\sin \left(2 x\right) = 2 \sin \left(x\right) \cos \left(x\right)$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = \frac{2 \sin \left(x\right) \cos \left(x\right)}{1}$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = \frac{2 \sin \left(x\right) \cos \left(x\right) {\sec}^{2} \left(x\right)}{\sec} ^ 2 \left(x\right)$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = \frac{2 \tan \left(x\right)}{\sec} ^ 2 \left(x\right)$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = \frac{2 \tan \left(x\right)}{1 + {\tan}^{2} \left(x\right)}$

Now let $\tan \left(x\right) = - \frac{3}{4}$

$\sin \left(2 x\right) = \frac{2 \left(- \frac{3}{4}\right)}{1 + {\left(- \frac{3}{4}\right)}^{2}}$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = - \frac{\frac{3}{2}}{1 + \frac{9}{16}}$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = - \frac{\frac{3}{2} \cdot 16}{16 + 9}$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = - \frac{3 \cdot 8}{25}$

$\textcolor{w h i t e}{\sin \left(2 x\right)} = - \frac{24}{25}$