[G.SRT.6] Similarity, Right Triangles, and Trigonometry #6

Objective

Common Core Text:

- [G.SRT.6] Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Said Differently:

- Understand the meaning of trigonometric ratios and how to use them

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Explanation

new vocab, opposite, adjacent, hypotenuse

Now that you understand similarity, we can start talking about an extremely useful part of math called Trigonometry. In high school Geometry, this means learning about Sine, Cosine, and Tangent ratios. These ratios only work for right triangles.

Let's see where these come from. Here are two right triangles

Remember,

a right triangle is a triangle that has a 90° angle.
- to establish similarity, a pair of triangles need 2 congruent angles (AA)

Since I already know that two right triangles have a congruent 90° angle, that means I just need 1 more congruent angle to prove that the two triangles are similar by AA.

So if I knew that ∠A ≅ ∠D then I would know that these two triangles are similar.

Well what do I learn if two triangles are similar? Their angles are congruent and their sides are proportional. Let's focus on the sides being proportional. That would mean that

Let's manipulate this a bit. If I multiply both sides by EF, I get

(EF) = (EF)

BC =

But then if I divide both sides by AB, I get

BC ÷ AB = ÷ AB

So what does that mean? Well, BC and EF are our opposite sides, while AB and DE are our adjacent sides. So this means that for any similar triangles, opposite divided by adjacent will be the same number.

This number, opposite divided by adjacent for an angle, is known as the tangent of an angle

Tangent of an Angle =

Remember for a right triangle, we only need to know one more angle to know the triangles are similar. That means any two right triangles with a congruent angle will have the same tangent.

This means we could pick an angle, say 1°, figure out the tangent of that angle, and copy it down into a table. Then, for the rest of eternity, whenever we saw an angle of 1°, we would already know the the ratio of opposite over adjacent.

Then we could do the same thing for 2°, then 3°, 4°,5°,6°,7°, all the way up to 90°, and we'd have a lot of useful information. And would you believe it! Mathematicians already did that.

But it gets better. Since all sides of similar triangles are proportionate, I could do the same thing for other ratios

Sine of an Angle =

Cosine of an Angle =

If you Google sine table and cosine table, you'll find tables like the tangent one above. Thus the birth of Sine (sin), Cosine (cos), and Tangent (tan). These are basically a way for us to get free information about the sides of triangles, just by knowing an angle. And that's useful

TL;DR

Because right triangles already have a right angle, we only need to have 1 more congruent angle to prove two right triangles are similar. That means for any right triangle, if we know another angle, then we automatically know the ratios of the sides. Mathematicians have named those ratios and use them all the time, because they allow us to get free information about sides of triangles just by knowing an angle.

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