Question

If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.

 

Answer 1

\[3 \sin x + 5 \cos x = 5 \left( Given \right)\]
Squaring both the sides: 
\[9 \sin^2 x + 25 \cos^2 x + 30 \sin x \cos x = 25\]
\[30 \sin x \cos x = 25 - 9 \sin^2 x - 25 \cos^2 x (1)\]
\[\text{ We have to find the value of 5 }\sin \theta - 3 \cos \theta . \]
\[ \left( 5 \sin x - 3 \cos x \right)^2 = 25 \sin^2 x + 9 \cos^2 x - 30 \sin x \cos x\]
\[ \left( 5 \sin x - 3 \cos x \right)^2 = 25 \sin^2 x + 9 \cos^2 x - \left( 25 - 9 \sin^2 x - 25 \cos^2 x \right) \left[\text{ From }(1) \right]\]
\[ \left( 5 \sin x - 3 \cos x \right)^2 = 34 \sin^2 x + 34 \cos^2 x - 25\]
\[ \left( 5 \sin x - 3 \cos x \right)^2 = 34 - 25 \left( \because \sin^2 x + \cos^2 x = 1 \right)\]
\[ \left( 5 \sin x - 3 \cos x \right)^2 = 9\]
\[5 \sin x - 3 \cos x = \pm 3\]