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Trigonometric Ratios and Similarity

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  • 1:45 Similarity
  • 3:07 Trigonometric Ratios
  • 6:03 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

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.

Right Triangles

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.

Right triangle with theta
Right triangle with angle beta

Similarity

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.

Two right triangles ABC and DEF with similar sides and same angles

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.

Trigonometric Ratios

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.

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