Solving Trig Equations: Finding Equilibrium & Period

[cscott]

Trig Functions

$$2y - 5 = \sin(144t - 45)$$

How can I find when the object is at equilibrum? I know it's when y = 0, but how do I solve from there? I tried arcsine but it gives me a domain error.

How can I find the minimum in between [itex]0 \le t \le 10[/tex]?

Is the period of oscillation 0.625 degrees?

 

[Doc Al]

What makes you think equilibrium is when y = 0? The midpoint of the motion will be when sin() = 0.

 

[Jameson]

What makes you think equilibrium is when y = 0? The midpoint of the motion will be when sin() = 0.

Also, if you solve explicitly for y, you'll see that there is a vertical shift, meaning that the y-axis is not the midpoint of this graph. Use Doc Al's advice.

As for the period, I didn't check your numbers, but remember in a sine graph in the form of $$a \sin{(bx+c)}+d$$ that $$\frac{2\pi}{|b|}$$ is equal to the period. That's in radians of course.

 

[cscott]

Alright, I revised my answers given the replies so far. I think the period is 2.5 degrees, maximum height (the question is about a spring oscillating) is 7.5m and the first equilibrum is at t = 0.3125. Can anyone tell me if I'm correct?

I'm still having trouble with the minimum in between [itex]0 \le t \le 10[/tex]?

 

[Doc Al]

The period should be in seconds, not degrees. (144 is in what units?) Rewrite your expression like this:
$$y = 2.5 + 0.5 \sin(144t - 45)$$

If you understand what this says, you should be able "read off" the equilibrium position, the amplitude, and the maximum and minimum values of y.

 

[cscott]

The period should be in seconds, not degrees. (144 is in what units?) Rewrite your expression like this:
$$y = 2.5 + 0.5 \sin(144t - 45)$$

If you understand what this says, you should be able "read off" the equilibrium position, the amplitude, and the maximum and minimum values of y.

Sorry, I meant at what times is the function at it's minimum between 0 <= t <= 10

As for the period, is it correct to say 2.5s instead of 2.5 degrees? I used what Jameson gave me: 360/|b| = T

I made the mistake of thinking the amplitude was 5 (no idea where I got that number, I've been juggling questions all night ;)... I see the max height should 3m, correct (assuming it's m vs t)?

 

[Doc Al]

Sorry, I meant at what times is the function at it's minimum between 0 <= t <= 10

y will be a minimum wherever sin() is at its minimum, which is when sin() = -1.

As for the period, is it correct to say 2.5s instead of 2.5 degrees? I used what Jameson gave me: 360/|b| = T

If the 144 is degrees/sec, then 2.5s is correct.

I made the mistake of thinking the amplitude was 5 (no idea where I got that number, I've been juggling questions all night ;)... I see the max height should 3m, correct (assuming it's m vs t)?

Right. Since the sin function oscillates between -1 and +1, y will oscillate between 2 and 3.

 

[cscott]

Thanks for your help!