Question

if 3 sin theta + 5 cos theta equals to 5 find 5 sin theta minus 3 cos theta​

Answer 1

Answer:

$\large\boxed{\sf{\pm3}}$

Explanation:

Let the angle theta is denoted by angle alpha.

Now, its given that,

$3 \sin( \alpha ) + 5 \cos( \alpha ) = 5 \: \: \: \: \: .......(1)$

To find the value of,

• $5 \sin( \alpha ) - 3 \cos( \alpha )$

Lets assume that,

$5 \sin( \alpha ) - 3 \cos( \alpha ) = x \: \: \: \: \: .........(2)$

Now, squaring and adding eqn (1) and (2),

$= > {(3 \sin( \alpha ) + 5 \cos( \alpha )) }^{2} + {(5 \sin( \alpha ) - 3 \cos( \alpha )) }^{2} = {5}^{2} + {x}^{2} \\ \\ = > (9 { \sin }^{2} \alpha + 30 \sin \alpha \cos \alpha + 25 { \cos }^{2} \alpha ) + (25 { \sin }^{2} \alpha - 30 \sin \alpha \cos\alpha + 9 { \cos }^{2} \alpha ) = 25 + {x}^{2} \\ \\ = > 34 { \sin}^{2} \alpha + 34 { \cos }^{2} \alpha = 25 + {x}^{2} \\ \\ = > 34( { \sin }^{2} \alpha + { \cos }^{2} \alpha ) = 25 + {x}^{2}$

But, we know that,

• ${ \sin }^{2} \gamma + { \cos }^{2} \gamma = 1$

Therefore, we will get,

$= > 34 \times 1 = 25 + {x}^{2} \\ \\ = > {x}^{2} = 34 - 25 \\ \\ = > {x}^{2} = 9 \\ \\ = > x = \pm3$

Hence, the required value is $\bold{\pm3}$