Working with Sines and Cosines

In some applications of calculus to physics and engineering, we have input information in the form:

Where is magnitude and is angular frequency, and we find that our mathematical analysis (somehow) results in the form

This is, in fact, still a sine wave. It has the same angular frequency as the input, but a different amplitude and phase. That is, it can be re-written in the form:

where is the amplitude, is the angular speed, is time, and is the phase shift.

This comes from the double angle identity for sine:

Example

Express in the form

Matching up the coefficients of the original statement with the new one:

To solve this system, we can square both equations and add the results

And with the Pythagorean identity, . Since is the amplitude, we take the positive root:

We can eliminate from the system by dividing the two equations. This gives by definition of tangent.

Reaching for the calculator, we find that radians. However, this is not necessarily the right value for . Since we selected to be positive, equations 1 and 2 tell us that is positive, and is negative, so must be in the 2nd quadrant, and not the 4th.

Working with Tangent.png

The angle is actually radians
We now have that

If is in the 1st or 4th quadrants, then , but if is in the 2nd or 3rd quadrants, then