SOHCAHTOA in the Unit Circle
- Whit Ford
Use this page to explore 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 (known as a Unit Circle) is convenient to use, as it ensures that all Hypotenuse measurements are equal to 1 (making denominators simple). Its equation is: Use the green slider below to alter the green angle α shown with its vertex at the center of the circle, and observe how the values in text at the top right change to reflect the angle you have chosen.
The green slider at the top left of the graph sets angle measures in "Radians", which are described as the length of the arc that the angle intercepts when in standard position on a Unit Circle. In this example, the arc length specified by the slider is arc AC on the Unit Circle. The Radian angle measure of a full circle is 2π, the circumference (arc length) 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" ratios: then fill in the measure you know, and solve for the one you seek. Verify you can convert between Radians and Degrees by doing this calculation yourself, and converting the Degree measure for α shown in the circle to Radians, then verifying that the number you obtains is 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 ratios. A vertical line is drawn from point C to the x-axis with the the point of intersection labeled B. B will always lie directly beneath C, and thus will always have the same x-coordinate as C. Note the right triangle OBC, with angle α at its bottom left corner and a right angle in its bottom right corner. Its right side, Opposite angle α, has a length equal to the y-coordinate of point C. The bottom of OBC, which is Adjacent to angle α, has a length equal to the x-coordinate of point C. The Hypotenuse of OBC is a radius of the Unit Circle, and therefore will 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 text above and to the right of the circle. Since we are in a circle with radius 1, OC will always have a measure of 1, and we can simplify the SOHCAHTOA ratios: 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 in the Unit Circle will always equal to the y-coordinate of the point that the angle intersects on the circle. By the same reasoning, Cos(α) = OB / OC = OB, and the Cosine of an angle in standard position in the Unit Circle will always equal to the x-coordinate of the point that the angle intersects on the circle. Click on the blue check box (labeled y-coordinate) above and to the left of 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. Click on the blue check box again to remove the check mark, then press ctrl-F on your keyboard to clear the Sine function graph from the screen. Click on the green 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: - When the α slider is at 0, 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 exactly what is being plotted on each axis. If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/