Extending Trigonometric Functions
- Author:
- Eric Karnowski
- Topic:
- Functions, Trigonometric Functions
1. Explore the unit circle. The unit circle, with center at (0, 0), has a point on it at (1, 0). Below it is a sliding point labeled “arclength = 0.” The arclength will be used as the input for the trigonometric functions.
a. Move the arclength slowly to the right to increase the arclength value. How does the point on the unit circle move?
b. What else do you notice about the circle as you increase the arclength?
c. What do you notice about the triangle?
d. The circumference of the unit circle is 2π. What happens when arclength > 2π?
e. What happens when you use a negative arclength?
2. Explore the sine function. Return arclength to 0, and then click the “Show function” checkbox. It should be set at “sine” already. Remember that sine is defined using the side opposite the angle. For the triangles you saw in the unit circle, that side is the height.
a. Move the arclength again. What happens to the height of the triangle? How does the red dot show this change?
Click the “Moving circle” box and move the arclength again. The unit circle will move so that the point on its edge stays at the same x-coordinate as the arclength used for the input. You may decide whether to keep the “moving circle” checked or not, unless your teacher instructs you otherwise.
b. What do you notice about the connection between the triangle and the curve graphed by the red point?
c. Take notes about anything you find interesting or surprising.
3. Explore the cosine function. Return arclength to 0, and then slide the point beside the “Show function” checkbox from “sine” to “cosine.” Remember that cosine is defined using the side adjacent to the angle. For the triangles you saw in the unit circle, that side is the base of the triangle.
a. Move the arclength again. What happens to the base of the triangle? How does the green dot show this change? Note: If you have the “moving circle” box checked, a second triangle (colored green) appears. This triangle is always equilateral, so that the height of this triangle—defining where the green dot is—is always the same length as the base.
b. What do you notice about the connection between the triangle and the curve graphed by the green point?
c. Take notes about anything you find interesting or surprising.
4. Explore the tangent function. Return arclength to 0, and then slide the point beside the “Show function” checkbox from “cosine” to “tangent.” Remember that tangent is defined using a ratio of the sides, opposite/adjacent.
a. Move the arclength again. What happens to the ratio height/base of the triangle? How does the blue dot show this change?
b. Take notes about anything you find interesting or surprising.