Question

If 5cos A=7sin A find sin A and cos A

Answer 1

Hello!

$\boxed{\sf 5\: cosA = \sf 7 \:sinA}$

$\textbf{Squaring\: both \:sides \:}$ :

25 cos²A = 49 sin²A

⇒ 49 sin²A = 25 (1 - sin²A) 

⇒ 49 sin²A +25 sin²A = 25

⇒ 74 sin²A = 25

⇒ sin²A = $\dfrac{\sf 25}{\sf 74}$

⇒ $\textbf{sinA}$ = $\dfrac{\sf 5}{\sqrt{74}}$

Therefore,

⇒ $\textbf{cosA}$ = $\dfrac{\sf 7}{\sqrt{74}}$

Then :

$\dfrac{\sf ( 7\:sinA + 5\:cos A )}{\sf ( 5\:sinA + 7\:cosA )}$

= $\dfrac{\sf 35/\sqrt{74} + \sf 35/\sqrt{74}}{\sf 25/\sqrt{74} + \sf 49/\sqrt{74}}$

= $\dfrac{\sf 70/\sqrt{74}}{\sf 74/\sqrt{74}}$

= $\boxed{\dfrac{\sf 35}{\sf 37}}$

Cheers!