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# The calculator gives me sine wave

Question. "When I put in 'sine' in my [graphics] calculator, it gives me sine wave.  Why?"

The sine of an angle in a right triangle is defined as the ratio between the opposite side and hypotenuse.  In the table below you see the values of sine for all even angles from 0 to 90 degrees.  Most of the time the sine of an angle ends up being an irrational number, so the values in the table are actually just rounded values to six decimal places.  Sometimes, though, the sine of an angle is not an irrational number but a rational number.  Can you spot a couple of special angles where the sine of the angle is a rational number?

 Angle Sine of angle Angle Sine of angle Angle Sine of angle 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 0.034899 0.069756 0.104528 0.139173 0.173648 0.207912 0.241922 0.275637 0.309017 0.34202 0.374607 0.406737 0.438371 0.469472 0.5 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 0.529919 0.559193 0.587785 0.615661 0.642788 0.669131 0.694658 0.71934 0.743145 0.766044 0.788011 0.809017 0.829038 0.848048 0.866025 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 0.882948 0.898794 0.913545 0.927184 0.939693 0.951057 0.961262 0.970296 0.978148 0.984808 0.990268 0.994522 0.997564 0.999391 1

This data gives us 46 pairs of numbers.  Each number pair corresponds to a single point on the coordinate plane.  In each case, the x-value is the angle, and the y-value is the sine of that angle.

This is how it looks like when those 46 dots are plotted.  It's the start of the familiar sine wave form!

This picture was done with Microsoft Excel, which most people have on their home PC.  What a great exploration tool Excel can be for studying plotting functions!  Just create x- and y-values in columns, highlight the area of the data, choose "Chart Wizard", and choose "XY-scatter plot".

If you plotted more points, the graph would look more continuous.  Or you could connect the existing points with lines.  That's exactly what a scientific calculator does: it calculates a certain number of points, and connects those with lines.

In this table are the same values for the sine of angle, and the same angles, but the angles are measured in radians, which is usually the default way of measuring angles in calculators.  The graph produced looks the same.

 Angle (radians) Sine of angle Angle (radians) Sine of angle Angle (radians) Sine of angle 0.000 0.035 0.070 0.105 0.140 0.175 0.209 0.244 0.279 0.314 0.349 0.384 0.419 0.454 0.489 0.524 0 0.034899 0.069756 0.104528 0.139173 0.173648 0.207912 0.241922 0.275637 0.309017 0.34202 0.374607 0.406737 0.438371 0.469472 0.5 0.559 0.593 0.628 0.663 0.698 0.733 0.768 0.803 0.838 0.873 0.908 0.942 0.977 1.012 1.047 0.529919 0.559193 0.587785 0.615661 0.642788 0.669131 0.694658 0.71934 0.743145 0.766044 0.788011 0.809017 0.829038 0.848048 0.866025 1.082 1.117 1.152 1.187 1.222 1.257 1.292 1.326 1.361 1.396 1.431 1.466 1.501 1.536 1.571 0.882948 0.898794 0.913545 0.927184 0.939693 0.951057 0.961262 0.970296 0.978148 0.984808 0.990268 0.994522 0.997564 0.999391 1

(Picture done with Microsoft Excel)

Now you might ask, "Doesn't the graph continue?  What about angles that are greater than or equal to 90 degrees?  I can't even draw a right triangle that would have a 91 degree angle (not even one with 90 degrees)."

You are right.  You can't draw a right triangle with a 90 or 91 or 200 degree angle.  That's where the right-triangle definition of sine comes to its limit, and that's where the unit circle and radians that you see in math books come into play.  With the unit circle, one can define sine and other trigonometric functions for all kinds of angles, and using that definition, one can then draw the graph of sine function further and get the familiar wave:

(Picture done with Microsoft Excel)