Answer to Question #216716 in Trigonometry for Unknown346307

Question

23.2 If f(t) = 2 sin t + 3 cost, express f(t) as a single sinusoid and hence determine its

amplitude and phase.

Answer 1

Solution:

$f(t) = 2 \sin t + 3 \cos t$

Let $f(t)=A \sin(t+\theta)$ , where $A$ is amplitude and $\theta$ is phase.

So, $2 \sin t + 3 \cos t=A \sin(t+\theta)$

$\Rightarrow2 \sin t + 3 \cos t=A [\sin t \cos\theta+\cos t \sin \theta] \\\Rightarrow2 \sin t + 3 \cos t=A \sin t \cos\theta+A\cos t \sin \theta$

On comparing both sides,

$A\cos\theta=2,A\sin\theta=3 \\\Rightarrow \tan\theta=\dfrac32 \\\Rightarrow \theta=56.31\degree$

Then, $A\cos\theta=2$

$\Rightarrow A\cos 56.31\degree=2 \\ \Rightarrow A=3.605$

Thus, amplitude is 3.605 and phase is $56.31\degree$ .