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

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 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

*y* = cos *x* has amplitude 1 and period

*y* = tan *x* has no amplitude and period

*y* = *a* sin b*x* has amplitude *a* and period

*y* = *a* cos b*x* has amplitude *a* and period

*y* = *a* sin b*x* has no amplitude and period

### Examples

1. Sketch in the domain for .

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2. Sketch for .

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will be moved units to the left. The graph is the same as 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 for .

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

4. Sketch for .

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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 in the domain ?