 In case it's been a while since you've done trigonometry, here's a quick review of the three basic trigonometric functions. Trigonometry begins with the study of right triangles, so let's take a right triangle apart. Suppose I take a right triangle and pick one of the acute angles. I can talk about the sides of the right triangle with respect to this acute angle. First, there's the hypotenuse, which is always the side across from wherever the right angle is. Notice that the hypotenuse forms one side of the acute angle. The other side of the acute angle is going to be called the adjacent side. Finally, there's the third side of the triangle, which is opposite from the acute angle. Now that I have these sides defined, it's helpful to consider three ratios between lengths of sides. The sine of the acute angle, that's the ratio, the quotient of the opposite side over the hypotenuse. There's the cosine of the angle, which is the ratio of the adjacent over the hypotenuse. And finally, there's the tangent of the angle, which is the ratio of the opposite side over the adjacent. For example, suppose we have a right triangle and one indicated angle, and we want to find the sine, cosine, and tangent of this angle. The first thing we'll want to do is we'll want to identify what our three sides are. So remember, the hypotenuse is always the side across from the right angle. The hypotenuse is one side of our angle, the other side is called the adjacent side. And finally, the side across from the angle is the opposite side. To find the sine, cosine, and tangent, let's go ahead and write down their definitions in terms of these sides. And notice that for cosine and for tangent, we have to know what the length of the adjacent side is. So we have to figure out that third side of a right triangle. So according to the Pythagorean theorem, the sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse. So our unknown side a squared plus 8 squared will equal 13 squared. We can then do a little bit of algebra to find the length of a. And now we have the length of the opposite, the adjacent, and the hypotenuse, so we can find all three trigonometric values. So sine is the opposite over the hypotenuse. That's 8 over 13. Cosine is the adjacent over the hypotenuse. That's square root 105 over 13. And finally, tangent is opposite over adjacent. That's 8 over square root 105. As another example, suppose we have a right triangle and we want to find the length of one of the sides of the right triangle. So here, as before, we want to identify what parts we actually have. So notice that we have an angle here, and a cross from that angle will be the opposite side, the adjacent side will be one of the sides of the angle, and the hypotenuse is always a cross from the right angle. Being able to identify the three sides allows us to decide which of the trigonometric functions we're going to use. Now we'll pull in our definitions of the three trigonometric ratios for reference, and we see that the side BC that we want is what we're calling the opposite side, and that the other length that we have is the adjacent side. So that means tangent is going to be the most useful. So we can set up our equation. We know that the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. The angle itself has a measure of 40 degrees, and the adjacent side has a length of 25. And so a little algebra tells us that the opposite side has a length 25 times tangent of 40 degrees, or about 20.975.