Explanation
Hint: Here, we have to apply trigonometric formulas
Solution:
Step1: Simplify the given displacement
$$x = 4\left( {cos\pi t{\text{ }} + {\text{ }}sin\pi t} \right).....(1)$$
Multiply & divide each term of R.H.S. of equation (1) by $$\sqrt 2 $$ we get,
$$ \Rightarrow \dfrac{4}{{\sqrt 2 }} \times \sqrt 2 \left( {cos\pi t{\text{ }} + sin\pi t} \right)$$
$$ \Rightarrow 4\sqrt 2 \left( {\dfrac{1}{{\sqrt 2 }}\cos \pi t + \dfrac{1}{{\sqrt 2 }}\sin \pi t} \right).....(2)$$
Step2: Apply required trigonometric formula we know that, $$\dfrac{1}{{\sqrt 2 }} = \sin \dfrac{\pi }{4}$$ and $$\dfrac{1}{{\sqrt 2 }} = \cos \dfrac{\pi }{4}$$
Now, equation (2) becomes,
$$ \Rightarrow 4\sqrt 2 \left( {\cos \pi t\sin \dfrac{\pi }{4} + \sin \pi t\cos \dfrac{\pi }{4}} \right).....(3)$$
Also, $$\sin (a + b) = \sin a\cos b + \cos a\sin b$$
Now, equation (3) becomes,
$$x = 4\sqrt 2 \sin \left( {\pi t{\text{ }} + \dfrac{\pi }{4}} \right).....(4)$$
Step3: Find Amplitude ‘A’
Comparing above equation (4) with,
$$x = A(\sin \omega t + \Phi )$$
We get, Amplitude, $$A = 4\sqrt 2$$
Hence, option (C) is correct.
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