Plot the corresponding reference circle for each of the following simple hoarmonic motions. Indicate the intial`(t=0)` position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case `: (x` is in cm and t is in s ).
(a) `x=-2 sin(3t+pi//3)` (b)`x=cos(pi//6-t)` (c) `x=3sin(2pit+pi//4)` (d) `x=2cospit`

Explanation

If we express each function in the form `x=acos(omegat+phi)` …(i)
where `phi` is the initial phase, i.e., `phi` represents the angle which the intial radius vector of the particle makes with the positive direction of x-axis.
(a) `x=-2sin(3t+pi//3)=2cos((pi)/(2)+3t+(pi)/(3)) or x=2cos(3t+(5pi)/(6))`
Comparing it with equation (i), we note that `a=2, omega=3 and phi=5 pi//6`
Hence, the reference circle will be as shwon in figure.
(b) ` x=cos((pi)/(6)-t)=cos(t-pi//6)` `[:' cos(-theta)=costheta]`
Comparing it with equation (i), we not that `a=1, omega=1 and phi=-pi//6.`
The reference circle will be as shown in figure.
`(c) x=3sin(2pit+pi//4)=3cos(2pit+(3pi)/(2)+(pi)/(4))`
Comparing it with equation (i), we note that `a=3, omega=2pi ` and ` phi=(3pi)/(2)+(pi)/(4)=(7pi)/(4)`
The reference circle will be as shown in figure.
(d) `x=2cos pit.`
Comparing it with equation (i), we note that `a=2, omega=pi` and `phi=0`.
The reference circle will be