# Discovering SOH-CAH-TOA

- Author:
- GeoGebra Materials Team

- Topic:
- Cosine, Sine, Tangent Line or Tangent

What do Sin, Cosine, and Tangent equal?

Follow the steps and answer the questions on a separate sheet of paper.
Hint #1: Check and uncheck boxes depending on which trigonometric function you are thinking about.
Hint #2: In steps 7-11, you may need to check the Tip #1 and Tip #2 boxes.
Step 1: Notice the right triangle within the quarter circle. What are the sides of a right triangle called?
Step 2: Move D along quarter circle. What happens to the hypotenuse? What happens to the legs of the triangle?
Step 3: Move D along quarter circle. What happens to α? What happens to sinα? cos α? tan α? Do you notice anything about the numerators and denominators or the sin, cos, and tan?
Step 4: Make a conjecture about what the sin α, cos α, and tan α are in relation to the triangle.
Step 5: Move point B to make quarter circle bigger or smaller? Test your conjecture from step 4 by moving D along new quarter circle. Does your conjecture seem to be true? If not make another conjecture and test that one.
Step 6: Compare your conjectures with a classmate.
Step 7: Move point B so that the radius of the quarter circle is 1 again. Move D along quarter circle so that α = 45°. What do you notice about sin α, cos α, and tan α when α = 45°.
Step 8: Move D along quarter circle so that α = 30°. What do you notice about sin α, cos α, and tan α when α = 30°.
Step 9: Move D along quarter circle so that α = 60°. What do you notice about sin α, cos α, and tan α when α = 60°.
Step 10: What conjectures can you make about the sin 45°, cos 45°, tan 45°? What conjectures can you make about the sin 30°, cos 30°, tan 30°? What conjectures can you make about the sin 60°, cos 60°, tan 60°?
Step 11: Are these conjectures always true? Test them by changing the radius of the quarter circle by moving point B.