Math 140

Class 12

Graph of Cosine

\(f(t) = a\cos\left(\omega t + \phi\right) + b\), \(\qquad g(t) = a\sin\left(\omega t + \phi\right) + b\)

Amplitude \({}= \left\lvert a\right\rvert\)
Period \({}= \frac{2\pi}{\omega}\)
Phase shift \({}= -\frac{\phi}{\omega}\)
Phase \({} = \phi\)
Vertical shift \({}=b\)
- baseline: \(x = b\) for cosine
- baseline: \(y = b\) for sine
Angular frequency \({} = \omega\)
Frequency \({} = \frac{\omega}{2\pi} = \frac{1}{\text{period}}\)

\(f(t) = a\tan\left(\omega t + \phi\right) + b\)

Amplitude \({}= \left\lvert a\right\rvert\) (distance from baseline at 1/4 period from phase shift)
Period \({}= \frac{\pi}{\omega}\)
Phase shift \({}= -\frac{\phi}{\omega}\)
Phase \({} = \phi\)
Vertical shift \({}=b\)
- baseline: \(s = b\)
Asymptotes at phase shift \(\pm\) 1/2 of period.

\(f(t) = \frac{1}{2}\tan\left(\frac{\pi}{3}t + \frac{\pi}{4}\right) + 2\)