Actually Applicable Application Problems and Brainteasers/Use a Unit Circle to Estimate Sine and Cosine

Overview

Sometimes you might need to know the sine or cosine value of a certain angle to complete a problem. There are some different ways to do this -- this method is one option.

General Method

1. Open up a GeoGebra tab (or app on your phone, etc.).
2. Plot the points (0,0) and (1,0).
3. Use the circle tool to make a circle centered on (0,0), through the point (1,0). This is the "unit circle."
4. Drop a point onto any other location on the unit circle.
• It should be possible to drag this point around the circle but not off of it. If that doesn't work, delete the point and try again.
5. Change this point's settings to show its coordinates.
6. Draw a line segment from (0,0) to the new point.
7. Use the angle measurement tool to label and measure the angle made by that segment and the x-axis.
• Notice that the angle display and measurement update automatically as you move the point around, as do its coordinates.
8. The draggable point's x-coordinate is the cosine of the labeled angle, and its y-coordinate is the sine of the labeled angle. If the angle whose sine and cosine you want is not the angle shown, drag the point until it is.

Problems

Evaluate sines

${\displaystyle \sin(90^{\circ })=}$

${\displaystyle \sin(37^{\circ })=}$

${\displaystyle \sin(120^{\circ })=}$

Evaluate cosines

${\displaystyle \cos(45^{\circ })=}$

${\displaystyle \cos(53^{\circ })=}$

${\displaystyle \cos(95^{\circ })=}$