INSTRUCTOR: Express the function x of t equal to cosine, parentheses, 4t, close parentheses, plus 4 sine, open parentheses, 4t, close parentheses, in the form A sine, open parentheses, omega times t plus phi, close parentheses. What is the frequency of motion? What is the amplitude?

So first, we can find our amplitude using the formula A equals the square root of c sub 1 squared plus c sub 2 squared, end square root. Now, looking at this equation, we have one cosine and four sines. So those are c1 and c2, respectively. So in fact, let me just write this c1 equals 1. c2 equals 4.

And also, because we have cosine, parentheses, 4t, close parentheses, and sine, open parentheses, 4t, close parentheses, that's our omega. Omega is equal to 4. The coefficient on the t is the argument inside cosine and sine, those two functions. All right. We're going to need that to find the frequency.

All right. So c1 equals 1. c2 equals 4. Omega equals 4. So let's plug this in and find the amplitude.

So the square root of 1 squared plus 4 squared, end square root. That'll be the square root of 17, end square root. So there's our amplitude.

Now, in order to find the frequency, that is omega divided by 2 pi. So our frequency equals omega divided by 2 pi. So this will be 4 as a numerator and a denominator of 2 pi, so that the frequency is 2 over pi.

Now, combining bits and pieces of this-- oh, actually, there's one more thing we need to find. We don't know what phi is. We're going to need that for our equation.

All right. So we know there's another fact. And that is tangent of phi equals c1 divided by c2, which means tangent of phi equals 1/4, or phi is equal to arctangent of 1/4, which is approximately 0.245. So we can write that x of t equals square root of 17, end square root, times sine, open parentheses, 4t plus 0.245, close parentheses.