Question

If 3sin.x + 4cos.x = 5, then find 4sin.x + 3cos.x

Answer 1

3sinx + 4cosx = 5
3sinx = 5 - 4cosx
9sin²x = 25 + 16cos²x - 40cosx
9 - 9cos²x = 25 + 16cos²x - 40cosx
25cos²x - 40cosx + 16 = 0
25cos²x - 20cosx - 20cosx + 16 = 0
5cosx(5cosx - 4) - 4(5cosx - 4) = 0
(5cosx - 4)(5cosx - 4) = 0

thus, cosx = 4/5 so. 3cosx = 12/5
substituting in given equation we get,
3sinx + 4(4/5) = 5
3sinx = 5 - 16/5
3sinx = 9/5
sinx = 3/5. so, 4sinx = 12/5

4sinx + 3cosx = 12/5 +12/5 = 24/5