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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)

Comments

How do you draw a sine wave/ cos wave graph for example, y = 2sin(x-pi)+1
Kay

I assume you mean plotting the graph by hand, and not by computer or calculator. The basics of graphing are explained above. After you can draw y = sin(x) by hand, then you need knowledge of these things:
a) How does the grahp of y = sin(x) compare with y = sin ( x - pi)
b) How does the graph of y = (something) compare with y = 2(something)
c) How does the grapy of y = (something) compare with the graph of y = (something ) + 1
Answers:
a) It is shifted to the right by Pi.

b) all y values are doubled. It's loke zooming in on y axis only in a sense. Or it becomes "elongated" in y axis sense.

c) the graph is 'lifted up" by one unit.

These pictures were done with MathGV, a free function plotting software.

Math Lessons menu

Place Value Ideas Teaching tens and ones Tens and ones practice Counting in groups of ten Skip-counting (0-100) Comparing (0-100) Cents and dimes Place Value - hundreds Comparing (0-1000) Place value - thousands Comparing thousands Rounding Place value - big numbers	Add/Subtract lessons Missing addend concept (0-10) Addition facts with 6 Addition & subtraction connection Within 0-100 Add/subtract whole tens (0-100) Easy principle of adding ones Fact families for basic facts Sum over next whole ten Add 2-digit numbers mentally Subtract 2-digit numbers mentally Carrying to tens Borrowing from tens Within 0-1000 Rounding to nearest ten Carry two times Borrow from hundreds Borrow two times Borrow over zero tens	Multiplication & Division Multiplication Multiplication concept as repeated addition Multiplication on number line Commutative Multiply by zero Word problems Order of operations Oral drilling guide Drilling tables of 2, 3, 5, or 10 Drilling tables of 4, 11, 9 Multiplying by whole tens Multiplying by 100 Distributive property Multiplying in columns the easy way Multiplying in columns Estimation when multiplying Division Division as making groups Division/multiplication connection Division is repeated subtraction Zero and one in division Not exact division/remainder Divisibility Long division as repeated subtraction Long division and why it works Remainder & long division Long division 0-10,000 Two-digit divisor Divisibility within 0-1000 Divisibility rules Factoring Sieve of Eratosthenes
Fraction Lessons Understanding fractions Part of whole group Mixed numbers Mixed number to fraction Fraction to mixed number Adding like fractions Equivalent fractions Adding unlike fractions Adding mixed numbers Subtracting mixed numbers Subtracting mixed numbers 2 Part of whole group 2 Measuring in inches Comparing fractions Simplifying fractions Whole number times a fraction Fraction times a fraction Multiplication as an area Simplify before multiplying Division of fractions - explain why it works Divide fractions by whole number Dividing fractions by fractions Dividing mixed numbers	Geometry Lessons Lines, rays, and angles Measuring angles Angles with the same vertex Parallel & perpendicular Triangles Altitude of a triangle Angles in a triangle Equilateral & isosceles triangles Circles Compass ruler constructions Symmetry Polygons Area of rectangles Area of right triangles Area of parallelograms Area of triangles	Decimals Lessons Tenths Tenths and place value Add decimals with tenths Multiply decimals with tenths Hundredths Hundredths as a place value Add decimals with hundredths Add decimals with hundredths 2 Multiply with hundredths Money problems. Comparing decimals

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