# Angular Displacement Differentiation

1. Jun 7, 2012

### richyr33

1. The problem statement, all variables and given/known data
An angular displacement θ radians in time t seconds is given by the equation θ = sin 3t. Find

a:) angular velocity when t = 1 second
b:) the smallest positive value of t for which the angular velocity is 2rad/s
c:) the angular acceleration when t = 0.5 seconds
d:) the smallest positive value of t for which the angular acceleration is 9rad/s

2. Relevant equations
d/dx (sin ax) = a cos ax
d/dx (cos ax) = -a sin ax

3. The attempt at a solution
Ok i have worked out parts a:) and c:)

a:) dθ/dt = 3cos 3t so when t = 1, ω = -2.97r/s

c:) dω/dt = -9sin 3t so when t = 0.5, angular acceleration = -8.98r/s/s

I can't work out parts b and d, its not a homework question i am revising from a book and the answers for parts b and d are 0.280seconds and 1.57 seconds respectively however the method is not explained. I managed to calculate the answer to part d at one point but cannot remember how i got there :shy:

Any help would be appreciated, thanks

2. Jun 7, 2012

### Muphrid

This really just boils down to saying, for example on part (b),

$$\dot \theta = 3 \cos 3t = 2$$

Move the 3 over so you get $\cos 3t = 2/3$ and then use an inverse cosine function. That's really all there is to it; (d) is very similar.

3. Jun 8, 2012

### richyr33

Ok thats great thanks!