If f(x)=3sinx+4sin(x-3) what is the amplitude and period (p) of f(x)?
No idea how to solve this.
No idea how to solve this.
To do problems like this, one way to do that would be rewrite the expression in the form of y = A sin (bx + c), so that you can immediately identify the period which would be \frac{2 \pi}{b} and the amplitude which would be A.
We will repeatedly use the formula \sin (A + B) = \sin A \cos B + \cos A \sin B to achieve this.
Let's first expand \sin (x - 3) = \sin x \cos 3 + \cos x \sin 3 .
Then f(x) = (3 + 4 \cos 3) \sin x + 4 \sin 3 \cos x .
Then f(x) = A \sin (x + B) where A = \sqrt{(3 + 4 \cos 3)^2 + 4 \sin^2 3} = \sqrt{9 + 4 \cos^2 3 + 24 \cos 3 + 4 \sin^2 3} = \sqrt{13 + 24 \cos 3} and B=\sin^{-1}\left(\frac{4\sin3}{A}\right).
Therefore, the period is 2 \pi and the amplitude is \sqrt{13 + 24 \cos 3} .
I hope this helps.