# If Cos X = − 3 5 and X Lies in the Iiird Quadrant, Find the Values of Cos X 2 , Sin X 2 , Sin 2 X . - Mathematics

Numerical

If $\cos x = - \frac{3}{5}$  and x lies in the IIIrd quadrant, find the values of $\cos\frac{x}{2}, \sin\frac{x}{2}, \sin 2x$ .

#### Solution

$\cos x = - \frac{3}{5}$ Using the identity
$\cos2\theta = \cos^2 \theta - \sin^2 \theta$ , we get
$cosx = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}$
$\Rightarrow - \frac{3}{5} = 2 \cos^2 \frac{x}{2} - 1$
$\Rightarrow 1 - \frac{3}{5} = 2 \cos^2 \frac{x}{2}$
$\Rightarrow \frac{2}{5} = 2 \cos^2 \frac{x}{2}$
$\Rightarrow \frac{1}{5} = \cos^2 \frac{x}{2}$
$\Rightarrow \cos\frac{x}{2} = \pm \sqrt{\frac{1}{5}}$
It is given that x lies in the third quadrant. This means that
$\frac{x}{2}$  lies in the second quadrant.
$\therefore \cos\frac{x}{2} = - \frac{1}{\sqrt{5}}$
Again,
$\text{ cos } x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}$
$\Rightarrow - \frac{3}{5} = \left( - \frac{1}{\sqrt{5}} \right)^2 - \sin^2 \frac{x}{2}$
$\Rightarrow - \frac{3}{5} = \frac{1}{5} - \sin^2 \frac{x}{2}$
$\Rightarrow - \frac{1}{5} - \frac{3}{5} = - \sin^2 \frac{x}{2}$
$\Rightarrow \frac{4}{5} = \sin^2 \frac{x}{2}$
$\Rightarrow \sin\frac{x}{2} = \pm \frac{2}{\sqrt{5}}$
It is given that lies in the third quadrant. This means that
$\frac{x}{2}$  lies in the second quadrant.
$\therefore \sin\frac{x}{2} = \frac{2}{\sqrt{5}}$
$Now,$
$\text{ sin } x = \sqrt{1 - \cos^2 x}$
$\Rightarrow \text{ sin } x = \sqrt{1 - \left( - \frac{3}{5} \right)}^2$
$\text{ sin } x = \sqrt{1 - \frac{9}{25}} = \pm \frac{4}{5}$
Since x lies in the third quadrant, sinx is negative.
$\therefore \text{ sin } x = - \frac{4}{5}$
$\Rightarrow \sin2x = 2\text{ sin } x\text{ cos } x$
$\Rightarrow \sin2x = 2 \times \left( - \frac{4}{5} \right) \times \left( - \frac{3}{5} \right)$
$\Rightarrow \sin2x = \frac{24}{25}$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.1 | Q 28.1 | Page 29