This graph and worksheet allows you to investigate how Right Triangle Trigonometry (SOH-CAH-TOA) can be used to derive all of the Sine and Cosine function values. A circle with Radius = 1 is convenient to use, as it ensures that all Hypotenuse measurements are equal to 1 (making denominators simple).
It may take a minute for the GeoGebra applet below to load, so please be patient (and give it permission to load if you receive any warnings from your browser).
Please scroll down and read through the text below the GeoGebra applet for some suggested tasks that will help you understand how the values of the Sine and Cosine functions are related to the Unit Circle:
x^2 + y^2 = 1
If you have comments or suggestions for this page, please click here to make them via my blog entry relating to these applets.
Note that the radius of the circle is 1.
1) Move the point along the green slider labelled α at the top of the applet and watch what happens.
Notice how this slider determines the measure of angle O, which can be positive (moving counter-clockwise) or negative (moving clockwise).
The slider is calibrated in Radians so that this example will fit in the window above. The "Radian measure of an angle" is the length of the arc that the angle intercepts when in standard position on a Unit Circle, which in this case is arc AC. The Radian angle measure of a full circle will be 2π, which equals the circumference of a Unit Circle.
The angle measure shown in the circle is in Degrees. If you wish to convert between Radian and Degree measure, you may do so by setting up two "Part to the Whole" proportions
Radians / 2π = Degrees / 360
then fill in the measure you know, and solve for the one you seek.
Verify you can convert between Radians and Degrees by checking that the Degree measure for α shown in the circle is correct, given the Radian measure shown above the α slider. You may wish to try this for several settings of the slider.
Remember to make sure you put Degrees over Degrees, and/or Radians over Radians, when using the above proportions.
2) A perpendicular line was drawn from point C to the x-axis, and the point where this line intercepts the x-axis is labelled B. The point B will always lie directly beneath C, and thus will share its x-coordinate.
Note the right triangle formed by OBC, with the angle α in its bottom left corner, and the right angle in its bottom right corner.
The right side, Opposite α, has a length equal to the y-coordinate of point C.
The bottom, Adjacent to α, has a length equal to the x-coordinate of point C.
And the Hypotenuse is a radius of the circle, and will thus always have a length of 1.
Note how these relationships are true no matter what angle measure you set for α.
We can now use Right Triangle Trigonometry (SOHCAHTOA) to determine the values of the Sine and Cosine functions of α, as shown in the blue text below the circle.
Since we are in a circle of radius 1, OC will always have a measure of 1, and we can simplify
Sin(α) = BC / OC
Sin(α) = BC / 1
Sin(α) = BC
So, Sin(α) will always equal the measure of BC, which is the y-coordinate of point C. So, the Sine of an angle in standard position is always equal to the y-coordinate of the point that the angle intercepts on the Unit Circle.
By the same reasoning, Cos(α) = OB / OC, will always equal the measure of OB, which is the x-coordinate of point C. So, the Cosine of an angle in standard position is always equal to the x-coordinate of the point that the angle intercepts on the Unit Circle.
3) Click on the top check box (y-coordinate) above the circle, just below the α slider, so that a check mark appears in the box.
Move the α slider left and right again. Note how the Point labelled "Sin(α)" is always at the same height as the point C (because Sin(α) = the y-coordinate of C), and note how it traces out one complete period of the Sine function as you move the α slider across its full width.
The Sine function has a starting value of zero (when α = 0, the y-coordinate of C is zero), rises to one, falls back to zero, continues falling until it gets to negative one, then rises back to zero before beginning the process all over again.
4) Click on the top check box again to remove the check mark, then press ctrl-F on your keyboard to clear the Sine function graph from the screen.
5) Click on the second check box (x-coordinate) above the circle, and move the slider around again. From the information given above, you should now be able to figure out:
- Why does Cos(α) start at 1?
- Sin(α) moved so nicely with point C... why isn't Cos(α) as synchronized? Hint: think about what is being plotted on the x-axis, and what is being plotted on the y-axis.