| -5π/4,3π/4 | -3π/2,π/2 | -7π/4,π/4 |
| -π,π | -2π,0,2π | |
| -3π/4,5π/4 | -π/2,3π/2 | -π/4,7π/4 |
Each position on the unit circle (the middle graph above) is an angle in radians. The right side point of the circle is 0 radians because that is where we start. Since the unit circle has a 1 radius that point also corresponds to an angle of 2π. If you go 2π around the circle you will be back to where you started. Going backwards all around the circle corresponds to an angle of -2π. The top of the circle is π/2 around the circle. If you touch or mouse the top of the circle you will see -3π/2 and π/2.
The cos of an angle is the x position of the angle on the unit circle. The sin of an angle is the y position of the angle on the unit circle. So an angle will be on the circle at point (cos,sin).
The altitude is added to the trignometric formula. It controls the center of the circle, if it is a cosine formula it controls the x position and if it is a sine formula it controls the y position. In the display below you can change the altitude and see how it changes the position of the circle. You can change between sin and cosine formulas. The amplitude changes the size of the circle. You can change the amplitude in the display below as well. You can also touch the graph on the left and see how the circle generates the cosine and sine functions.
| Formula | Circle | Graph |
|---|---|---|
The frequency controls how fast the circle turns. The period is defined as how far along the graph it takes to complete a cycle. You can measure the period on a graph by the distance between peaks (or tops) of the curve. Using bottoms works too. Frequency is related to period using the equation: frequency*period = 2π. You can see the effect of frequency by changing the frequency in the display above and clicking the animate button.
The display below contains a random red trigonometry function. Try to enter the formula for that red function below. If you get it right it will give you another problem. If you get it wrong it will display your function in blue.
You can also play this in game mode by hitting start game. If you get the function right on the first try you will get 3 points. If it takes 2 tries you get 2 points and so on. You have 3 minutes to get the most points you can.