Hey everyone. So in a recent video, we talked about the special 45, 45, 90 right triangles, and we learned how you can use some shortcuts to quickly find the missing sides. Well, in this video, we're going to be learning about the common trig functions and how they are associated with the 45, 45, 90 triangle. And we're going to see if we can recognize any kinds of patterns for these trig functions, because what we're going to learn for these special cases is there may be shortcuts when finding ratios for the trigonometric functions as well, meaning these triangles could be overall a lot easier to solve. So I'm always interested in finding shortcuts, and hopefully, you are too. So let's get right into this.

Now we're going to start with the sine function, and we know that sine is the opposite divided by the hypotenuse. Now if we go to a 45 45 90 right triangle, we can go to either one of the two angles because they're both 45 degrees. So if we take a look at this angle, for example, and we go to the opposite side, we can see that the opposite is 5. And if we divide this by the hypotenuse or long side, that's going to be 5 times the square root of 2. We can cancel the fives here, giving us 1 over the square root of 2. And by rationalizing this denominator, I'll multiply the top and bottom by the square root of 2. This will get the square roots to cancel, giving us square root 2 over 2. So that means for the sine, we'll end up with √2/2. But now let's try solving for the cosine. For the cosine, we get adjacent divided by hypotenuse. Now if we go to one of our 45 degree angles, the adjacent side is 5. And if we divide this by the hypotenuse, we'll have 5 times the square root of 2. The fives will cancel, giving us 1 over the square root of 2. We already figured out this is the same thing as square root 2 over 2, so this will just simplify to radical 2 over 2, which is the cosine for this right triangle.

Now let's take a look at the tangent. For the tangent, we have opposite over adjacent. So what I'm going to do is go over here to our right triangle, look at one of our angles. And if I go to the opposite side, we end up with 5. If I go to the adjacent side, we also have 5. So we'll have 5 over 5, which is simply 1, meaning the tangent for our 45, 45, 90 triangle is 1.

So let's see if we notice any patterns with these trigonometric functions. Well, something that I notice is that the sine and cosine are the same, and the tangent just comes out to 1. So this is something that's pretty straightforward about these triangles is that you get the same sine and cosine values. But let's take a look at these other reciprocal identities like the cosecant, secant, and cotangent. So for the cosecant, we know that this is just one over the sine of theta, which means all we're going to do is flip this fraction that we got. So we'll have 2 over the square root of 2, and then we can rationalize this denominator. When doing this, the square roots will cancel on bottom of the fraction, giving us 2 times the square root of 2 over 2. These are going to cancel right here, meaning all we're going to end up with is the square root of 2 for the cosecant. And because the sine and cosine were the same, that means one over the cosine, which is our secant, is also going to be the square root of 2.

So all we now have to solve for is the cotangent, which is just one over the tangent, but we learned that the tangent is 1. So the cotangent will be 1 over 1, which is simply equal to 1. So notice how really straightforward all these trigonometric functions are when you're looking at a special case 45, 45, 90 triangle. We can recognize that the sine and cosine will always be square root 2 over 2, that the cosecant and secant will always just be the square root of 2, and that the tangent and cotangent will always be 1. These are all of the common trig functions for the 45, 45, 90 triangle. Hope you found this video helpful, and thanks for watching.