Question

for problems 4 and 5 identify the correct equation for the plot (the answers are under the plot to which they refer). recall that the function ( x(t)=acos(omega t) ) has an amplitude ( a ), and a period of ( t = 2pi/omega ).
4 (5pts) a) ( cos(pi t / 2) ) b) ( cos(4pi t) ) c) ( cos(t) ) d) ( cos(2pi t) ) e) ( cos(pi t / 4) ) f) ( cos(2t) )

Explanation:

Step1: Determine the period formula

The period formula for \(x(t) = A\cos(\omega t)\) is \(T=\frac{2\pi}{\omega}\).

Step2: Analyze each option

• Option a: For \(x(t)=\cos(\frac{\pi t}{2})\), \(\omega=\frac{\pi}{2}\). Then \(T = \frac{2\pi}{\frac{\pi}{2}}=4\).
• Option b: For \(x(t)=\cos(4\pi t)\), \(\omega = 4\pi\). Then \(T=\frac{2\pi}{4\pi}=\frac{1}{2}\).
• Option c: For \(x(t)=\cos(t)\), \(\omega=1\). Then \(T = 2\pi\).
• Option d: For \(x(t)=\cos(2\pi t)\), \(\omega = 2\pi\). Then \(T=\frac{2\pi}{2\pi}=1\).
• Option e: For \(x(t)=\cos(\frac{\pi t}{4})\), \(\omega=\frac{\pi}{4}\). Then \(T=\frac{2\pi}{\frac{\pi}{4}} = 8\).
• Option f: For \(x(t)=\cos(2t)\), \(\omega=2\). Then \(T=\frac{2\pi}{2}=\pi\).

From the graph, when \(t = 1\), the function completes one full - cycle.

Answer:

d) \(\cos(2\pi t)\)