Sine Function - Graph Exercise

Sine wave curve plotted from 0 to 360 degrees

The Sine Function produces a very beautiful curve,
but don't take our word for it, make your own!

Sine Function

First, read the page on Sine, Cosine and Tangent.

Now you will know that the sine of any angle is the length of the far side of the triangle (the "opposite") divided by the long side (the "hypotenuse"):

Right triangle with angle theta, opposite side, adjacent side, and hypotenuse
Sine of θ = Opposite / Hypotenuse

Draw Triangles

To make the graph, we need to calculate the sine for different angles, then put those points on a graph, and then "join the dots".

Step 1: Draw the Angled Lines

Place a mark at the center of a piece of paper, then, using a protractor, mark every 15 degrees from 0° to 180° in a semi-circle. Then rotate the protractor and mark from 180° around to the start again. Then draw lines radiating from the center to each of your marks so that you end up with an illustration like this:

Circle with radiating lines spaced every 15 degrees

Lines at 15° (click to enlarge)

Or, you can click on the above illustration, then print out the result.

Step 2: Draw and Measure the Triangles

From the end of each angled line, draw a line straight down (or up) at right angles to the horizontal center line. This forms a right triangle for each angle:

Right triangle formed by dropping a vertical line from the angled spoke to the horizontal axis

Measure Triangles

When you have completed each triangle, it is simply a matter of measuring the lines. Remember that the sine is the length of the line opposite the angle divided by the hypotenuse (which should all be the same length if you have drawn it well)

Right triangle with hypotenuse and vertical opposite side highlighted for measurement

Write all your measurements in a table. This is what I got, but your measurements may be different:

Angle	Opposite	Hypotenuse	Opposite / Hypotenuse
0°	0 mm	86 mm	0.00
15°	22 mm	86 mm	0.26
30°	43 mm	86 mm	0.50
and so on ...

You can print a table ready to fill in here.

When the "opposite" line goes downwards it is negative.

Tip: if you have drawn it well, you can take advantage of the symmetry of 0−90, 90−180, 180−270 and 270−360.

Graph The Results

Get some graph paper and prepare it by scaling off 0 to 360 in 15 increments along the x-axis, and scaling off −1 to +1 on the y-axis. You can use your own graph paper, or print out this graph paper

Now plot each point from the table on the graph.

Then join the dots as neatly as you can.

Result

The result should look something like the graph at the very top.

But you have done much more than draw a nice curve. You have :

learned about one of the most important functions in mathematics
learned that you don't have to believe what people say - you can try it for yourself
had experience plotting graphs
learned how symmetry can save effort

We hope you had a great time exploring!