Where do terms like sine, cosine and tangent come from? In this lesson, we'll learn about how similarity with right triangles leads to trigonometric ratios.
In triangle world, triangles come in all shapes and sizes. It's just like it is with people. You know how people from the same family share certain characteristics? For example, I've gotten recognized by distant cousins because we share the same large nose. Yeah, we're all thrilled about that. Anyway, triangle families also share certain characteristics.
Today, we're going to talk about the right triangle family. A right triangle is just a triangle with one right angle. So, all triangles from this family share that one right angle, like a nose that can't be missed.
There's more to right triangles, though. Like a family that dresses in matching outfits, right triangles have sides we can label just because they come from the same family. The side across from the right angle is called the hypotenuse. This is always the longest side, since it's across from the biggest angle.
These other two sides have names that change depending on which angle you're focusing on. It's kind of like how if you wear pants on your head, you call it a hat. Okay, not really. Anyway, let's say that we want to talk about this right triangle in terms of this angle here, which we'll call theta.
So, that's one member of our right triangle family. What if we also look at his older sister? Here's ABC's sister DEF.
Plus, look at the lengths of the sides of ABC. AC is 3, BC is 4 and AB (the hypotenuse) is 5. With DEF, we see that DF is 6, EF is 8 and DE (the hypotenuse) is 10. ABC and DEF are similar. That means they have the same shape, just not the same size. DEF is a few years older than ABC, hence the size difference.
The sides of similar triangles are in proportion to one another. This is important. Consider the ratio in ABC of the side opposite theta to the hypotenuse. That's 3/5. In DEF, the radio of the corresponding sides is 6/10. 3/5 = 6/10.
We could do the same with the opposite to the adjacent sides. In ABC, it's 3/4. In DEF, it's 6/8. 3/4 = 6/8.
So, similar triangles have sides in proportion to one another. What does that have to do with trigonometry? Everything! This fact about right triangles leads us to trigonometric ratios. A trigonometric ratio is a ratio between two sides of a right triangle.
You've heard of sine, cosine and tangent? All these terms do is put labels on these ratios of similar triangles we've been talking about. It's like finding out there's a name for whatever it is those Kardashian sisters share.
That first ratio we looked at, the opposite over the hypotenuse? We call that sine. We can say sin(theta) = opposite/hypotenuse.
Then there's cosine. We can define this as the adjacent over the hypotenuse. So, cos(theta) = adjacent/hypotenuse.
If we extend our family metaphor to these trigonometric ratios, sine and cosine are maybe like twins. You know those people who name their twins something cute, like Jaden and Kaden or Faith and Hope? That's your sine and cosine. Note that they both feature the hypotenuse on the bottom of the ratio.
And then there's tangent. Tangent is the opposite over the adjacent. So tan(theta) = opposite/adjacent. That's like the third sibling who feels left out sometimes, but goes on to have a fulfilling life making geometry lesson videos. Whoa, got a little personal there.
Anyway, that's all that sine, cosine and tangent mean. If a right triangle has an angle theta (which just stands in for any angle), any triangle with that same angle will be similar. Why? Because if two right triangles share the angle theta, then they're similar. Remember, if two angles of a triangle are equal, so is the third angle. And if the three angles are the same, then the triangles are similar - same angles, just maybe different sides, but the sides are still in proportion.
And so the sine of theta, or the ratio of the sides, will always be the same. If theta is 30 degrees, then that ratio of the opposite over the hypotenuse will have one value. If theta is 45 degrees, it'll have a different value. But for all triangles with that 30 degree angle, sine of theta is always the same. It's like all of the Kennedy family having that same accent. Well, except Arnold Schwarzenegger, but he married in.
We can remember these terms with the acronym SOH CAH TOA. That's sine = opposite/hypotenuse, cosine = adjacent/hypotenuse and tangent = opposite/adjacent. SOH CAH TOA. Some people mix up the first two and try to say SAH COH TOA. Remember, there's that internal rhyme to this acronym: so - cah - toe - ah. Os before As.
To summarize, we looked at the right triangle family. These triangles all share that right angle. When their angles are all equal, then they're pretty much siblings. They're also called similar. This similarity leads us to the trigonometric ratios, which are ratios between two sides in a right triangle.
First, there's sine. This equals the opposite over the hypotenuse. Then there's cosine, which is the adjacent over the hypotenuse. Finally, there's tangent, which is the opposite over the adjacent. We can remember these ratios with the acronym SOH CAH TOA. Remember, it rhymes: so - cah - toe - ah.
At the end of this lesson you should be able to:
- Recall what makes two right triangles similar
- Calculate sine, cosine and tangent
- Recall the helpful hint SOH CAH TOA