This right angled triangle.

This is a right triangle.

It is equal to 90 degrees.

So there I have a right triangle.

Let's call this theta.

let's say, let me draw another right triangle, that's another right triangle here.

Well, you could imagine a world where I told you that this is a right triangle.

But this forms a right triangle.

So this is called a right triangle.

This right triangle looks like this, it looks like this, this is, this angle is sixty minus theta, and this height over here is four

And what that told us is, let's say we had a right triangle.

And now I'm going to draw a right triangle again.

I always like to do it, so let's just draw a right triangle over here.

We know this is a right triangle, et cetera.

And how do we know that it's the only right triangle that has this property?

So now we're looking at this large right triangle here.

Our opposite is a 2, the hypotenuse is equal to r.

But let's say that's a right triangle, and let's call that theta.

But this definition, using a right triangle like this, actually breaks down at certain points. Actually, at a lot of points.

So this right triangle right over here I could draw like this, I could draw like this.

So what's interesting about a 45 45 90 triangle is that it is a right triangle that has this property.

Well, similarly, if we look at this right triangle, because this is also a right triangle, of course, x here, relative to alpha, is also the opposite side, and B now is the hypotenuse.

So, we have an angle of 90 , and thus a rectangular triangle.

So this side over here is maybe 3. This height right over there is 3.

If I would tell you the length of this side is 5, and if our tell you that this angle is 45 degrees, can we figure out the other two sides of this triangle?

Triangle ABC is a right triangle with AC 7 BC 24 and a right angle at C. So let's just try to draw that.

So if we have a triangle, and the triangle has to be a right triangle, which means that one of the three angles in the triangle have to be 90 degrees.

But maybe we can use this extra information right here, this 45 degrees, to figure out another side, and then we'd be able use the Pythagorean theorem.

The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle.

So let's say that we have we want to figure out what cosine of let's say, negative a is.

So if I were to give you a right triangle, and I were to tell you that this angle is 90 and this angle is 45, and that this side is, let's say this side is 8.

So if this is a right triangle, then we could have used the Pythagorean theorem somehow, but now we can't.

And to break it down into its horizontal and vertical components, we just set up a right triangle and just use our basic trig ratios.

Because if you think about it, this is a right triangle. This is a right triangle. That has distance 3.

This right here is a right angle.

Or x is equal to 45 degrees. So the only right triangle in which the other two angles are equal is a 45 45 90 triangle.

Rectangular

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Also, the center of the circle that circumscribes a right triangle is the midpoint of the hypotenuse and its radius is one half the length of the hypotenuse.

Well, if we look at this larger right triangle, that is the opposite of beta times its hypotenuse.

Geometric Right Angle Triangle

Rectangular

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If we look at this right triangle right here, is opposite over hypotenuse for alpha. So it's sine of alpha.