 Section 8.4 is all about trigonometry. We'll use trigonometry to solve problems about right triangles, and we'll use it in two ways. The first way is we'll use it to find missing side lengths of right triangles, and then secondly we'll use inverse trigonometry to find angle measures. Before we begin, let's review some vocabulary. Whenever you have a right triangle, the side opposite the 90 degree angle is called the hypotenuse. The other two legs have specific names based on your frame of reference. For example, if we're standing down at angle A, the leg that's opposite us is literally called the opposite leg. The leg that's next to us is the adjacent leg, and it's important to keep that distinction in mind when dealing with trigonometry. Trigonometry is a way of relating these side lengths back to that original angle of reference. So for example, if I asked for the sine of angle A, the sine is defined as opposite divided by hypotenuse. The cosine of angle A is the ratio adjacent over hypotenuse, and the tangent of A is the ratio of opposite over adjacent. Now it's critical for you to remember these relationships, and so the mnemonic soka toa is oftentimes used to help you remember these. Sine is opposite over hypotenuse, cosine is adjacent divided by hypotenuse, and the tangent function opposite over adjacent. So let's try an example. Here we're asked for the sine of L, the cosine of L, and the tangent of L, and that means L is our angle of reference. So that means 8 refers to the opposite, 15 is adjacent, and hypotenuse is 17. So the sine of angle L, opposite over hypotenuse, eight-seventeenths. Cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Let's change this around. Now I'm asked for the sine of N, which means I'm now at angle N as my angle of reference. So the sine of angle N is opposite over hypotenuse, opposite now is 15-seventeenths. Cosine is adjacent over hypotenuse, and tangent, wait a minute, that tangent is wrong. What should it be? Mmm, opposite over adjacent. It should be 15 over 18. Oops, sorry about that. So to find a missing side length with trigonometry, first you need a right triangle with a given angle and a given side length. Next you set up a trig ratio equation. In this case we would use the sine function and then solve for the unknown x. I'll walk you through an example. So we have that triangle from the previous slide. Your angle of reference is the angle that you know, in other words 26 degrees. From that angle of reference we know we want to solve for the opposite and we know the hypotenuse. And so that means the sine function is for us. So the sine of the angle equals opposite over hypotenuse. So there's our equation. Now we need to solve for x. If you have equations in that same general format like 5 equals x over 8, the way that we solve for x is by multiplying by 8 on both sides. And so we'll do the same thing for our trig equation. If it's something equals x over 43, we'll multiply both sides by 43 to get a final solved equation. x is 43 times the sine of 26 degrees. And then it's time to pull out a calculator. 43 times the sine of 26 should give us 18.85. If instead you have a calculator that is like this one, you'll have to type in 43 and then hit the multiply button times. And then type in 26 first, then tap sine, then tap equals 18.85. So why does this work? The reason trigonometry works out is each right triangle with a 26 degree angle, they're all similar to each other. And so the ratios of the sides are always going to be the same. So we divide our hypotenuse will be equal in small triangles, big triangles, so long as they have a 90 degree angle and a 26 degree angle. Let's try another example then. Here we have a 40 degree angle, 25 units side and x. So our angle of reference up top there. Why is that the angle of reference? The reason that's the angle of reference is it's the only angle that we know that isn't the 90 degree angle. In this case we know the opposite. We want to solve for the adjacent leg. We'll use the tangent function. So the tangent of the angle is opposite over adjacent. Now this one looks a little bit different. In the previous example we had x divided by a number. Now we have a number divided by x. So just note that whenever you have a number divided by x, if you multiply both sides by x, you'll then have to divide. So the tangent of 40 equals 25 over x. That's the same as 25 divided by the tangent of 40. And so then your calculator will give you approximately 29.79. Likewise with this calculator 25 divided by and then type 40, then type tangent, then press equal. One more example. Let's take a look at this one. Here we've got a 60 degree angle x and 38. So we know the hypotenuse. We want to solve for the adjacent. That's the cosine function. So the cosine of 60 degrees is x over 38. Solving for x gives us 38 times the cosine of 60. And we get exactly 19 units. Wait a minute. If we have a 60 degree angle in a right triangle, that means the other angle has to be 30 degrees. And so we have a special right triangle. So of course we would have the 2n side equals 38, which means n has to be 19. One final example. Try it on your own. Here we've got a 70 degree angle, a 20 unit side, and x. Pause the video. Give it a shot on your own. You should have used the tangent function. So the tangent of the angle equals opposite over adjacent. So therefore x equals 20 times the tangent of 70, which is about 54.95 units long. The next video will deal with finding an angle measure using trigonometry.