# Trigonometric Graphs

The values of Unit Circle in Trigonometry create a repetitive pattern producing graphs like waves. These graphs are written in domains of $\pi$.

The three basic trigonometric functions: sine, cosine and tangent display different graphs at different points.

The Geogebra applet below shows their pathway between 0 and 2$\pi$ radians. "Click to play" the sin, cos and tan graphs.

Furthermore, the inverse trigonometric functions also have their own set of graphs. The graphs are reciprocal ratios of the basic sin, cos and tan graphs (and values).

### Examples:

1. Graph the function f(x)=sin(x)+1

Notice that the normal sine graph moves up by one point. The range has a minimum of 0 and a maximum of 2.

2. Graph the function f(x)=2cos(x).

Notice that the normal cosine graph's range has been doubled in this function. The "2" doubles the range.

3. Graph the function f(x)=sin(2x).

Notice that there are double the amount of waves compared to the normal sine graph. "2x" increases (or decreases) the frequency (cycle) of the wave.

## Trigonometric Functions with Periods and Amplitudes

The period is the distance over which the curve moves along the x-axis before it repeats. While the amplitude is the maximum distance that the graph stretches out from the centre of the graph on the y-axis.

y = sin x has amplitude 1 and period $2\pi$

y = cos x has amplitude 1 and period $2\pi$

y = tan x has no amplitude and period $\pi$

y = a sin bx has amplitude a and period $\cfrac{2 \pi}{b}$

y = a cos bx has amplitude a and period $\cfrac{2 \pi}{b}$

y = a sin bx has no amplitude and period $\cfrac{\pi}{b}$

### Examples

1. Sketch $y=5\sin { 4x }$ in the domain for  $0 \le x \le 2\pi$.

2. Sketch $f(x)=\sin { \left( x+\cfrac { \pi }{ 2 } \right) }$ for  $0 \le x \le 2\pi$.

$f(x)=\sin { \left( x+\cfrac { \pi }{ 2 } \right) }$ will be moved $\cfrac{\pi}{2}$ units to the left. The graph is the same as $f(x)=\sin {x}$ but starts in a different position.

If you are not sure, you can always use tables of values (although it takes a longer time).

3. Sketch $f(x)=\cos { \cfrac { 1 }{ 2 } x }$ for  $0 \le x \le 2\pi$.

Notice that the period is a half - which means that the wave of cosine function will be longer and less frequent.

4. Sketch $y=\cos { 2x } +2\cos { x }$ for  $0 \le x \le 2\pi$.

The results of cos 2x and 2 cos x are joined together and the resultant values are used. To be more accurate, you can use a table of values.

Some trigonometric equations are not solvable algebraically, but it could solved graphically by finding the points of intersection between two equations. One example as follows:

How many solutions are there for $\tan { 2x } =\cfrac { 1 }{ 2 } x+1$ in the domain $0\le x \le 2\pi$?

Sketch  $y=\tan{ 2x }$  and  $y=\cfrac { 1 }{ 2 } x+1$.
We can see that $y=\tan{ 2x }$ and $y=\cfrac { 1 }{ 2 } x+1$ intersect at four different points. Therefore, the equation has 4 unique solutions.