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.
 * 5) * 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.
 * 6) Change this point's settings to show its coordinates.
 * 7) Draw a line segment from (0,0) to the new point.
 * 8) Use the angle measurement tool to label and measure the angle made by that segment and the x-axis.
 * 9) * Notice that the angle display and measurement update automatically as you move the point around, as do its coordinates.
 * 10) 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.

Evaluate sines
$$\sin(90^\circ)=$$

$$\sin(37^\circ)=$$

$$\sin(120^\circ)=$$

Evaluate cosines
$$\cos(45^\circ)=$$

$$\cos(53^\circ)=$$

$$\cos(95^\circ)=$$

Make Your Own Problem
Choose an angle for a particular purpose, like a rotation transformation, and evaluate its sine and cosine.