Every advanced algebra student deals with it eventually, not to mention calculus and trigonometry, where it comes from. The sinusoid is a curvy wave that has roots in triangle sides and angles, and it repeats itself over and over. That doesn’t mean it can’t look good!

Our Tools: The Waves

Sine functions come in the form y = a*sin(bx+c)+d, which I’ll break down for you here:

• y, for non-math people, is the height of any point on the coordinate grid. If you see an equation with y and x in it, it might relate to a set of points on the grid that fit that equation.
• a is the “amplitude” of our sine function, or (half of) how tall it is. For the graphic above, a = 1, which means that the top is one unit above an imaginary line drawn down the middle, and the bottom is one unit below this line. The total height is a + a or twice the amplitude.
• b relates to the “period” and “frequency” of the graph. Although it’s not one-to-one, the more b is, the more “squished” the graph will appear. (You would see more waves in a specific part, so if b went up in the graph shown above, there would be more bumps.)
• c relates to the “phase shift”. It’s not quite self-explanatory because the math people chose a weird phrase to use, but changing c moves the graph left and right.
• d, as you can tell, is just tacked on at the end as something that is added to it. It doesn’t make it taller, but the effect is something like if you took the entire thing and dragged it up a few units (or down a few, if d is negative).

But math doesn’t stop there! Although we could talk about the cosine wave (which looks like cos() in math), it’s the same thing as the sine above, but moved to the left. Trigonometry is better than two things that are basically the same, though. We have two other patterns at our disposal!

Despite the fact that it doesn’t exist in some places 😕, it’s actually another repeating, triangle-based function. As with our sine and cosine, you can stretch it and move it using math. As with our sine and cosine, there exists a copycat called the cotangent (it literally is just this pattern backwards)

The final relative of the two (all the inverse trig functions give up too quickly, so they don’t count) is the secant/cosecant duo. They look like the above image, and also are stretchable and movable with simple arithmetic. So let’s look at how these are created.

Where Did They Come From?

Here is a Pythagorean triple triangle, which means that it has a right angle and all the sides are whole numbers. Very clean! Now, let’s say we want to know how big the black angle is. Or we want to know the red angle but we don’t know the length of two of the sides. These are trigonometry problems! Although we don’t need to go in-depth too much here, this is how sine/cosine/tangent/the rest fit in.

For right triangles, the following are true:

• The sine of an angle, say the red one, is the side opposite it divided by the long side (the hypotenuse). This is 8/10 or 4/5.
• The cosine of the red angle is the side adjacent to it (that is not the hypotenuse) divided by the long side. This is 6/10 or 3/5.
• The tangent is the opposite divided by the adjacent, or for the red angle 8/6 or 4/3.
• We can derive the rest from the three listed above. The secant is 1 divided by the cosine, the cosecant is 1 divided by the sine, and the cotangent is 1 divided by the tangent.

Say we squish the triangle so the red angle is very small, close to zero. Then the opposite side is very close to zero. But that’s the numerator of the sine function, you might say! It is! That’s why, as angles get close to zero, sine gets close to zero. We get a number close to zero. Similarly, a large red angle means the sine is close to one, because it’s the reverse scenario. With stuff like this, we get the properties of the functions.

Now, we’ve also come up with a lot of equations that all of these formulas follow. These allow us to solve for missing angles or sides and can help when doing a geometry-intensive activity such as building something.

Anyway, that’s it for now! I hope you learned something new from this! And if you didn’t, here’s something that you might be able to say you learned from this: The Claddagh ring is a traditional Irish ring given which represents love, loyalty, and friendship. Done!