Question

If 5 cos theta + 7 sin theta is equal to 7 prove that 5 sin theta minus 7 cos theta is equal to plus minus 5​.

Answer 1

Step-by-step explanation:

$5 \cos(x) + 7 \sin(x) = 7 = >$

Squaring on both sides, we get

$25 {( \cos(x))}^{2} + 49 {( \sin(x))}^{2} +$

$2(5 \cos(x))(7 \sin(x)) = 49...(1)$

Let's take the value of 5sin(x)-7cos(x) be y.

Squaring on both sides ,we get

$25 {( \sin(x))}^{2} + 49 {( \cos(x))}^{2} -$

$2(5 \sin(x))(7 \cos(x)) = {y}^{2} ...(2)$

Add equations (1) and (2), we get

$74( {( \sin(x))}^{2} + { (\cos(x))}^{2} = 49 + {y}^2$

$= > 49 + {y}^{2} = 74 = > {y}^{2} = 74 - 49$

$= 25 = > {y}^{2} - 25 = 0 = >$

$(y - 5)(y + 5) = 0 = > y - 5 = 0 \: or \:$

$y + 5 = 0 = > y = 5 \: or \: - 5$

Hence the value of 5sin(x) -7cos(x) is equal to -5/+5. Hope it helps you.