What you know about triangles?! Sine, cosine, and tangent.

Trigonometry was such a little bastard for me coming up in high school. I took Geometry as a freshman, then Algebra 2/Trig, then Pre-Calculus and Calculus. To be honest I don’t even know what the fuck was taught in the “Algebra 2” section of the class (goddamn teacher handed out worksheets and just sat at his desk all period), but I remember trig being shoehorned into the class in a super awkward way. Then in the Calc classes we barely touched it except for graphing a few trig functions. I assume this is a common experience, as I’ve worked with tons of students who see trig as this inscrutable thing. Like, it’s fucking triangles? And there’s also this circle? The hell is Sohcahtoa? Fortunately, it’s easy motherfucking peasy as long as you can do a bit of memorizing or have easy access to a unit circle diagram. Let’s get it.

What is trigonometry?

It’s triangles. Measuring angles and sides and shit. In Greek it literally means “triangle measure.” Hard stop.

Why is it useful?

Because there are triangles in the world. And sometimes we need to know the lengths of their sides and the measures of their angles.

Say your goofy ass is tying a zip line from a tree to the ground. If you know how high on the tree you want it to be, and you know how far along the ground you want it to go, you can use trigonometry to figure out how much zip line you need. Or if you already have a certain amount of zip line because your lazy ass didn’t measure first, you can use trigonometry to determine the angle you need to hang it so that it reaches the ground.

You know, useful shit like that.

The ratios: sine, cosine, and tangent

These little homies are your basics. They take in an angle measure, and give you the ratios of the sides of a right triangle (a triangle that has a 90° angle). Easier to show than tell:

Here we have a triangle. It’s a right triangle, which we know because that box on the bottom right indicates that that angle is 90°. The other angle we need to understand is $\Theta$. $\Theta$ (pronounced “theta”) is the angle that we feed sine, cosine, and tangent to get the ratio of the sides. We could call this angle anything… $x$, $\Omega$, and Lebron are perfectly acceptable names for it, but it’s tradition to use $\Theta$. And it makes you feel like a damn smartypants.

The sides are termed “adjacent,” “opposite,” and “hypotenuse.” Hypotenuse is the side opposite the right angle. Simple. The adjacent side is the leg of the triangle that makes up $\Theta$. The opposite side is the side opposite $\Theta$. It does not touch $\Theta$ at all.

Now that we understand the parts of a triangle, we can start bringing in these ratios.

Ratio definitions

Sine is the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side. In math speak that’s

$$\sin(\Theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}$$

$$\cos(\Theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}$$

$$\tan(\Theta)=\dfrac{\text{opposite}}{\text{adjacent}}$$

This is where that SOH CAH TOA shit comes from. Sin Opposite Hypotenuse, etc.

So in practice, if we know the sides of our triangle, we can calculate the sine, cosine, and tangent ratios of any of the angles. Take a triangle like this:

Since we know all the sides, we can calculate all the ratios for $\Theta$:

$$\sin(\Theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}=\dfrac{3}{5}=0.6$$

$$\cos(\Theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}=\dfrac{4}{5}=0.8$$

$$\tan(\Theta)=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{3}{4}=0.75$$

So far, this is baby shit. Just doing fractions and saying that’s what these ratios are equal to. But if you know a little bit more of yo shit you can actually determine the measure of $\Theta$.

Let’s say for the sake of argument that you have memorized in that bright little head of yours that the sine ratio of 30° is 0.5. Don’t worry about how you know it for now, just trust me that it is. Let’s say you come across this little beauty of a triangle here:

So from this you can determine the following

$$\sin(\Theta)=\dfrac{3}{6}=\dfrac{1}{2}=0.5$$

That’s all well and good, but then you think to yourself “Shit son, sine of 30° is 0.5! My boy on the ‘net told me!” So now all of a sudden you figured out that $\Theta$ is 30°.

And that is what is powerful about these ratios. They allow you to piece together a couple bits of information and determine a whole shitload about the triangle as a whole. Not only that, but these ratios hold no matter how big your triangles are. It doesn’t matter if it’s 3 inches long or 100 miles long, as long as they have the same angle (30° for example), the ratios remain the same.

Next up I’ll be diving into the absolute unit himself: the unit circle. There we’ll be able to figure out how to calculate the ratio of any angle and you’ll figure out why sine of 30° is indeed 0.5.

And then you can just use your calculator.

Peace.