Question

f(x)= 3sinx-5cosx as a single sine function which has been shifted horizontally and stretched vertically.

Answer 1

Try working backwards:
\[A\cdot \sin(x+y)= A\sin(x)\cos(y)+A\sin(y)\cos(x) = C_1\sin(x)+C_2\cos(x) \]

Answer 2

Where
\[C_1 = A\cos(y)\]
\[C_2 = A\sin(y)\]

Given C1 and C2, how can you find A and y?

Answer 3

let 3=r cosa and 5 =r sina now squaring we have r^2 =9+25 =34 hence r =sqrt(34)
(note r here is always taken +ve)
puttin it in f(x)= 3sinx-5cosx
we have f(x)= r cosasinx-r sinacosx =r sin(x-a) =sqrt(34)sin (x-a) where tana =5/3