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

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## Overview[edit | edit source]

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[edit | edit source]

- Open up a GeoGebra tab (or app on your phone, etc.).
- Plot the points (0,0) and (1,0).
- Use the circle tool to make a circle centered on (0,0), through the point (1,0). This is the "unit circle."
- 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.

- Change this point's settings to show its coordinates.
- Draw a line segment from (0,0) to the new point.
- 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.

- 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[edit | edit source]

### Evaluate sines[edit | edit source]

### Evaluate cosines[edit | edit source]

### Make Your Own Problem[edit | edit source]

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