The Sine Function

The graph of y = sin x (in degrees):

We can see how this graph is related to the angular position on the circle:

The graph on the left is the unit circle, and the graph on the right is a plot of sin θ  vs. θ  in rectangular coordinates. As you watch the animation, recall that the sine of an angle is the value of the y-coordinate of the unit radius vector on the polar graph. On both graphs, the sine of the angle is measured along the y-axis, and is represented by the dotted blue line (the horizontal black line serves no purpose other than to highlight the projection onto the y-axis).

The amplitude of the graph is the maximum vertical displacement (on the y-axis) of the graph. For the simple sine function, the amplitude is 1 because it is never more that one unit away from the x-axis.

The pure math definition of the period is the length along the x-axis of one whole cycle (a cycle is one complete repetition of the graph). For the simple sine function, the period is 360° (or 2π if the angular units are radians). This definition can be a little confusing, because when we talk about a sinusoidal function as a function of time, the period is the amount of time that one cycle takes. This is how it is used in practice, so when you see period, think time.

The General Sine Function as a Function of Time

The general sine function is used to represent a sinusoidal function with any amplitude and period as well as with a shift left or right (which is called a phase shift). It is usually used to represent some quantity such as an AC voltage that varies in time. In that case, the horizontal axis would be measured in time units such as milliseconds or microseconds, and the vertical axis would measure volts.

In this formula the independent variable is the time t. The other values (A, ω) would be specified. It would then give you the y-value (typically the voltage) at any time t.

·       In this formula it is understood that the sine function is calculated in radians, not degrees. Always make sure that your calculator is in radian mode when using this formula.

Amplitude: The amplitude is the maximum value of the function, which for an AC voltage is the peak voltage, Vp. Because the maximum value of the sine function is 1, multiplying it by any number causes that number to be the new maximum value. For example, household AC has a peak voltage of about 170 V, so the coefficient A would be 170.

Another way to describe the sine wave when the independent variable is time is by its frequency. The frequency is a measurement of cycles per second. The official metric unit for a cycle per second is called a Hertz (Hz). However, the argument (input) of the sine function must be an angle, and so it needs to be in units of either degrees or radians. The convention in AC electronics is to use radians, so the angular frequency ω is just the frequency measured in radians per second instead of cycles per second. Think of it as a unit conversion, to convert cycles into radians. Because there are 2π radians in a cycle,


Period: The period T is the length of time it takes to complete one cycle. For example, the period of ordinary 60-Hz household AC current is 16.67 ms (which is 1/60 of a second). The period is a measurement of  seconds per cycle, whereas the frequency is cycles per second, so it is reasonable that they are reciprocals of each other:



Disregarding phase, the equation for household AC voltage as a function of time would be

v(t) = 170 sin(377 t)

170 is the amplitude (the peak voltage), and 377 is simply 2πf with f = 60Hz.
With this formula we can find the voltage at any particular time.

For example, at t = 3 ms, v(3 ms) = 170 sin(377 (3 × 103)) = 154 V

(your calculator must be in radian mode)

Phase Angle: The phase angle  is the angular displacement (in either degrees or radians) by which the entire sine function has been shifted left or rightfrom its usual position. Notice that the graph of the simple sine function, y= sin x, passes through the origin. On the graph of the general sine function, though, it does not have to. The entire function is shifted left or right by a distance . You need to notice one thing about the signs: the minus sign means that it is shifted to the right. This is the opposite of what you might expect, because ordinarily right is the positive direction. For example, in a pure capacitance the voltage is phase-shifted 90° behind the current (it lags the current). Time moves from left to right on the horizontal axis, so this means that the graph of the voltage is shifted 90° to the right of the current, so the phase angle is 90°.

Phase Displacement: The phase  isplacement is the actual amount that the function is shifted in whatever units the time axis is using. In other words, it is the amount of time by which  he sine wave leads or lags the reference (unshifted) sine wave. The phase angle, on the other hand, is how many degrees of a cycle, based on the idea that one whole cycle I  always 360°, regardless of what units are used on the time axis. For example, suppose that the period of a sine wave is100 ms, and it is displaced by 50 ms. This corresponds to half of the period, so the phase angle is 180°, but the displacement is 50 ms. To convert from phase shift to phase angle, simply set up a proportion where the shift is a fraction of a whole cycle, in time units on one side and in angle units on the other side:


If an AC signal oscillates at 150 kHz with a 90° phase shift between the voltage and the current, what is the time delay between the current peak and the voltage peak?

First we need to know the period. Given that the frequency is 150 kHz, we can find the period from

P = 1/f = 1/(150 × 103 s) = 6.67 microseconds

The phase angle is 90°, which is one-fourth of a cycle. The period of 6.67 microseconds is the time for one full cycle, so the time for the 90° delay will be one-fourth of a period. Using the ratio formula:


Cross-multiply and solve for d to get 1.67 μs

Note: At this point we have a small item of units confusion to mention. The angular frequency ω = 2πf discussed above has 2π radians representing a full cycle, not 360°. For some reason, though, the phase angle is usually expressed in degrees, not radians. You just have to get used to this mixed use of angular units, and pay close attention at all times to which angular units are being used. Obviously, you cannot add radians and degrees together in the sine function–it must be all one or the other. This means that you may have to do some converting before you calculate values.