Time Period,Frequency and Phase of Trigonometric Sine and Cosine Functions
The Trigonometric Sine and Cosine Functions are also called Sinusoids.
The Sinusoids are Periodic Fucntions, i.e the Output Values of these functions Repeat/Cycle Back after a certain Interval of Increase or Decrease in its Input Values.
This Interval is also known as the Time Period. The Sinusoids have a Default Time Period of \(2\pi\).
The Reciprocal of Time Period of Sinusoids (or any Periodic Function) is called Frequency.
The Sinusoids have a Default Frequency of \(\frac{1}{2\pi}\).
Sinusoids can be made to have Any Arbitrary Frequency and Time Period by changing their Input values as follows
Time Period for \(\sin(\theta)\) (or \(\cos(\theta)\)) \(=2\pi\), Frequency =\(\frac{1}{2\pi}\)
Time Period for \(\sin(2\theta)\) (or \(\cos(2\theta)\)) \(=\frac{2\pi}{2}=\pi\), Frequency =\(\frac{2}{2\pi}=\frac{1}{\pi}\)
Hence, Time Period for \(\sin(N\theta)\) (or \(\cos(N\theta)\)) \(=\frac{2\pi}{N}\), Frequency =\(\frac{N}{2\pi}\)
And Time Period for \(\sin(2\pi \theta)\) (or \(\cos(2\pi \theta)\)) \(=\frac{2\pi}{2\pi}=1\), Frequency =\(\frac{2\pi }{2\pi}=1\)
\(\therefore\) Sinusoid for any Arbitrary Time Period \(T\) can be given by \(\sin(\frac{2\pi}{T}\theta)\) or \(\cos(\frac{2\pi}{T}\theta)\) which shall have the Frequency of \(\frac{1}{T}\)
And Sinusoid for any Arbitrary Frequency \(f\) can be given by \(\sin(2\pi f\theta)\) or \(\cos(2\pi f\theta)\) which shall have the Time Period of \(\frac{1}{f}\)
Phase of a Sinusoid is Any Angular Displacement that either gets Added to or Subtracted from the Input of the Sinusoid.
The Difference between Angular Displacement of any 2 Sinusoids (or any Periodic Function) of a Similar Frequency and Time Period is called the Phase Difference.
For example, the Trigonometric Sine and the Cosine functions have a Phase Difference of \(\frac{\pi}{2}\). That is,