Trigonometry review

Lesson 5W

To analyze oscillations, it is very helpful to be able to combine multiple sine and cosine terms into a single term. You can do this, as long as the sines and cosines you want to combine have the same frequency. Here is how it works:

Try it out

Write \(\sin(4t)-\cos(4t)\) in the form \(A\cos(4t-\delta)\).

Then use your work to calculate when \(e^t\sin(4t) - e^t\cos(4t)\) crosses the equilibrium position for the third time.

Animations

Here are links to animations to help you see how the solution changes as you change parts of the differential equation.

If you increase the damping while holding everything else constant, the quasi period increases.

If you increase the initial velocity (or initial position) while leaving everything else the same, you get the same basic oscillation shape, but at a higher amplitude.