Circular (Trig) Functions

One of the most used functions in physics, engineering or any real application of math is the trigonometric functions. They show up everywhere…

### Trig Function: Where'd they come from?

The trig functions or circular function come from the unit circle. Remember that the unit circle allowed us new definitions of the basic trig functions.

(1)
\begin{align} sin(\theta) = y \end{align}
(2)
\begin{align} cos(\theta)=x \end{align}

Imagine increasing the angle $\theta$ by small amounts and recording the corresponding x and y values on the unit circle. From this data you could generate a table of data that could then be graphed.

Now here's the tricky part - especially if you always think of the horizontal axis as the "x-axis" or the vertical axis as the "y-axis" - we are going to create two new graphs. On each of the graphs the horizontal axis will be the angle. On the first graph the vertical axis is the x values (cosine) from the unit circle this gives the graph:

On the second graph the vertical axis is the y-values (sine) from the unit circle which gives the graph:

Note that in both graphs the parts in red are the first "period" or the first positive revolution around the circle. The rest of the graphs come from making more than one revolution around the circle or going backwards (negative angle).

You should be very familiar with these graphs. To the point that you can draw the basic shape with max, mins and zeros in the correct positions from memory!

### "Animation" of the Creation of the Trig Functions

A quick and dirty Geogebra Applet showing how the trig function come about.

### Anatomy of Sine and Cosine Functions

Domain: All real numbers.
Range: $-1 \geq y \geq 1$ that is if the amplitude is 1

The trig function (because of its wide use) has special names for its characteristics.

Amplitude - Half the height of the function. Or the height from the Sinusoidal Axis to a max or min.

Period - the horizontal distance taken to repeat itself. Or the horizontal distance from one max (min) to the next max (min).

Sinusoidal Axis - An imaginary axis in the "middle" of the function.

Phase Displacement - Essentially the amount the function is shifted horizontally. Note that this is different for the sine and cosine function. For the cosine function phase displacement is (often most easily) measured from the vertical axis to the first max. For the sine function the phase displacement is measure from the vertical axis to the first time the function crosses the sinusoidal axis and has a positive slope.

### Transformations of Trig Functions

Just like any function trig functions can under go transformations. The transformations are the same as for any other function, but in some cases the parameters that are doing the transformations have special names when talking about trig functions.

Writing the sine and cosine functions with all the parameters in them we get:

(3)
$$f(x)=A sin(B(x-C))+D$$
(4)
$$f(x)=A cos(B(x-C))+D$$

A represents a vertical dilation. It is called the "amplitude" of a trig function.

B is a horizontal dilation. IT IS NOT THE PERIOD, but is related to the period.

(5)
\begin{align} Period = \frac{2 \pi}{B}=\frac{360^\circ}{B} \end{align}

C is a horizontal shift. Sometimes call a phase displacement or phase shift (physics).

D is a vertical shift. Effectively this is the sinusoidal axis - so called because it's the "center line" of the function.

### The Less Talked About Tangent Function

The tangent function while useful, does not show up a lot in IB Math SL. It looks something like this:

Take special notice of the vertical asymptotes that occure every 180 degrees or every $\pi$ radians. They asymptotes occur when the cosine function is zero! This limits the domain of the tangent function. The range of the tangent function is all real numbers - there is no max or min value for the tangent function.

While rare to see, the tangent function can also under go transformations that behave just like all other transformations.

(6)
$$f(x)=A tan(B(x-C))+D$$

Amplitude is basically meaningless for the tangent function. The period of the tangent when $B=1$ is 180 degrees or $\pi$ radians.