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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 \[\cos\frac{x}{2}, \sin\frac{x}{2}, \sin 2x\] .

 

 

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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
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