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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 rounded to six decimal places. Sometimes, though, the sine of an angle is not irrational, but is a rational number. Can you spot a couple of such special angles where the sine of the angle is a rational number?

 Angle Sine of angle Angle Sine of angle 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 0 0.034899 0.069756 0.104528 0.139173 0.173648 0.207912 0.241922 0.275637 0.309017 0.342020 0.374607 0.406737 0.438371 0.469472 0.5 0.529919 0.559193 0.587785 0.615661 0.642788 0.669131 0.694658 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 0.71934 0.743145 0.766044 0.788011 0.809017 0.829038 0.848048 0.866025 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!

This picture was done with Microsoft Excel, which most people have on their home PC.  What a great exploration tool Excel can be for 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 plot more points, the graph would look more continuous. You could connect the existing points with lines — and that's exactly what a scientific calculator does: it calculates a certain number of points, and connects those with lines.

This table shows us the same things as the table above: angles and the sine of each angle, but the angles are measured in radians (not degrees). The graph produced looks the same.

 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 0.559 0.593 0.628 0.663 0.698 0.733 0.768 0 0.034899 0.069756 0.104528 0.139173 0.173648 0.207912 0.241922 0.275637 0.309017 0.342020 0.374607 0.406737 0.438371 0.469472 0.5 0.529919 0.559193 0.587785 0.615661 0.642788 0.669131 0.694658 0.803 0.838 0.873 0.908 0.942 0.977 1.012 1.047 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 0.71934 0.743145 0.766044 0.788011 0.809017 0.829038 0.848048 0.866025 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

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."

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

### See also:

How does a computer or calculator find values of sine?

### Comments

Question. How do you draw a sine wave/ cos wave graph?
For example, y = 2sin(x − Pi) + 1 ?

Kay

Hi Kay,
I assume you mean plotting the graph by hand, and not by a computer or a calculator. The basics of graphing are explained above. After you can draw y = sin(x) by hand, then you need to know these things:
1. How does the grahp of y = sin(x) compare with y = sin(x − Pi)?

2. How does the graph of y = (something) compare with y = 2(something)?

3. How does the graph of y = (something) compare with the graph of y = (something) + 1 ?

Once you understand (a), (b), and (c), you can also plot y = 2sin(x − Pi) + 1.

Answers:

a) It is shifted to the right by Pi units.

b) all y-values are doubled.

c) the graph is moved upwards by one unit.

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