1.

If \(\tan x = \dfrac{-3}{4}\) and \(\dfrac{3\pi}{2}< x < 2\pi\), then the value of sin 2x is1. 7/252. -7/253. 24/254. -24/25

Answer» Correct Answer - Option 4 : -24/25

Concept:

\(\rm \tan ^2x+1=sec^2x\)

tan x = sin x /cos x

sin 2x = 2 sin x cos x

 

The table below shows the sign of trigonometric ratios in different quadrants:

T – Ratio’s

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Sin

+

+

-

-

Cos

+

-

-

+

Cosec

+

+

-

-

Sec

+

-

-

+

Tan

+

-

+

-

Cot

+

-

+

-

 

Calculation:

Here, \(\tan x = \dfrac{-3}{4}\)

Squaring and adding 1 to both the sides, we get 

\(\rm \tan ^2x+1=(-\frac 3 4)^2+1\\ sec^2x=\frac {25}{16}\\ sec x=\pm\frac54\)

x is in fourth quadrant so, sec x= 5/4 and cos x = 4/5                       ......(\(\dfrac{3\pi}{2}< x < 2\pi\))

Now

 \(\rm \frac {\sin x}{\cos x}=\frac{-3}{4}\\ sin x= \frac{-3}{4}\times \frac{4}{5}=\frac{-3}{5}\)

Now, sin 2x  = 2sin x cos x 

\(=2\times \frac{-3}{5}\times \frac{4}{5}\\ =\frac{-24}{25}\)

Hence, option (4) is correct.


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