Explain an equation.

Bikerman2019

In an equation that says 5cos(9t)


5 cos is the pk amplitude cutting the Y axis, but what exactly is the 9t. Is is cycles per second?


Should know this but head a bit wonky today


Thanks

Michael Collins

Hi bikerman.

In 5cos(9t), the 5 is known as the 'peak amplitude' (often just called the 'amplitude'), while the 9t is known as the 'phase angle' (in radians).

This is not quite the same as cycles per second (also known as 'frequency')

The phase angle is required to change by 2*pi radians for the cos function to repeat i.e. go through one cycle.

Let's see what the number of cycles per second is (i.e. the frequency):

The variable t usually represents time, so as time increases, the phase angle increases.

Starting from t = 0 seconds, we have

5cos(9 x 0) = 5cos(0) = 5.

When 9t = 2*pi, the cos(.) function has gone through 2*pi radians, and we're back at

5cos(2*pi) = 5cos(2*pi) = 5 ... since cos(2*pi) = 1

If 9t = 2*pi, this means that t = 2*pi/9 seconds.

So, for one cycle, we must wait 2*pi/9 seconds. This is known as the period, usually given by T i.e. T = 2*pi/9.

How many cycles can be have per second? We need to see how many times we can divide 2*pi/9 seconds into 1 cycle:

1 cycle/(2*pi/9 seconds) = 9/(2*pi) cycles/second, this is the frequency, usually given the symbol f i.e. f = 9/(2*pi) cycles/second.

Another name for cycles per second is hertz, or Hz i.e. f = 9/(2*pi) Hz.

Note how this is a general rule i.e. f = 1/T, or, in words, frequency = 1/period.