Question

if 3 sin theta + 4 cos theta =5 then what is 4 sin theta -3 cos theta

Answer 1

Answer:

$\bold\red{4 \sin \theta - 3 \cos \theta = ±3}$

Step-by-step explanation:

For simplicity,

Let's denote 'theta' as 'alpha'.

Given,

$3 \sin( \alpha ) + 4 \cos( \alpha ) = 5$ .............(i)

We have to find the value of,

$4 \sin( \alpha ) - 3 \cos ( \alpha )$

Let,

$4 \sin( \alpha ) - 3 \cos( \alpha )=x$ .............(ii)

Now,

squaring and adding eqn (i) and (ii),

we get,

$= > (9 + 16) { \sin }^{2} \alpha + (16 + 9) { \cos }^{2} \alpha = 16 + {x}^{2} \\ \\ = > 25( { \sin}^{2} \alpha + { \cos}^{2} \alpha ) = 16 + {x}^{2}$

But,

we know that,

${ \sin }^{2} \alpha + { \cos }^{2} \alpha = 1$

Therefore,

putting the value,

we get,

$= > 16 + {x}^{2} = 25 \\ \\ = > {x}^{2} = 25 - 16 \\ \\ = > {x}^{2} = 9 \\ \\ = > x = \sqrt{9} \\ \\ = > x = ±3$

Therefore,

$4 \sin( \alpha ) - 3 \cos( \alpha ) = ±3$

Hence,

$\bold{4 \sin \theta - 3 \cos \theta = ±3}$