# How do you find sin 2x, given tan x = -2 and cos x > 0?

Nov 30, 2015

Find sin 2x, knowing tan x = -2, and cos x > 0

Ans: $\sin 2 x = \frac{4}{5}$

#### Explanation:

3 Trig identities to be used:
$1 + {\tan}^{2} x = \frac{1}{\cos} ^ 2 x$(1)
${\sin}^{2} x + {\cos}^{2} x = 1$ (2)
$\sin 2 x = 2 \sin x . \cos x$ (3)
Given tan x = -2. First find cos x and sin x
(1) --> $1 + 4 = \frac{1}{{\cos}^{2} x}$ --> ${\cos}^{2} x = \frac{1}{5}$ --> $\cos x = \pm \frac{1}{\sqrt{5}}$.
Since cos x > 0, then $\cos x = \frac{1}{\sqrt{5.}}$
(2) --> ${\sin}^{2} x = 1 - {\cos}^{2} x = 1 - \frac{1}{5} = \frac{4}{5}$ -->$\sin x = \pm \frac{2}{\sqrt{5.}}$
Since cos x > 0 then $\sin x = \frac{2}{\sqrt{5}}$.
(3) --> $\sin 2 x = 2 \sin x . \cos x = 2 \left(\frac{1}{\sqrt{5}}\right) \left(\frac{2}{\sqrt{5}}\right) = \frac{4}{5}$