 So let's introduce the three fundamental trigonometric ratios known as sine, cosine, and tangent. So one of the reasons that mathematics is useful is that in the real universe, there are some things that are easy to measure and some things that are hard to measure. Mathematics allows us to compute the hard things from the easy things. For example, acceleration is easy to measure, velocity is hard to measure. And we can use calculus to compute velocity from acceleration. Now calculus is a little bit further down the path of mathematics, but for right now angles are easy to measure and distance is hard to measure, and we can use trigonometry to compute distances from angles. Ultimately trigonometry will apply to any triangle, but we'll begin with a right triangle. So let's draw a right triangle. Not that, not that either. So remember, a right triangle has a right angle and two other angles. There we go, where the square marks where the right angle is. Now if we choose one of these other angles, we can identify the remaining sides. We have the hypotenuse, which is opposite the right angle. We have the leg adjacent to the chosen angle, and the leg opposite the chosen angle. And it's very important to remember that the opposite and adjacent sides are determined by where the angle is located. So now we can define three important trigonometric ratios. Given a right triangle and an acute angle with measure A, we define the trigonometric ratio sine, the length of the opposite side divided by the hypotenuse, cosine, the length of the adjacent side divided by the hypotenuse, and tangent, the length of the opposite side divided by the length of the adjacent side. And what that means is that given any right triangle and the length of all three sides, we can find the sine, cosine, and tangent of either of the acute angles in the triangle. So let's try to find the sine, cosine, and tangent of A in the triangle shown. Pulling in our definition, we have the sine opposite over hypotenuse, the cosine adjacent over hypotenuse, and tangent opposite over adjacent. Since we don't know the hypotenuse, we should find the value. Because this is a right triangle, we know that we can use the Pythagorean theorem in a right triangle with legs having length A, B, and hypotenuse C. We know that A squared plus B squared equals C squared. Now it's conventional to name the side after the angle it's across from. So this side here, across from angle A, we'll call that little a. This side here, across from angle B, we'll call that little b. And the hypotenuse is C. So that tells us little a is 3, little b is 7, and C is the hypotenuse which we don't know, but we can apply the Pythagorean theorem, and we find that the hypotenuse has length square root of 58. And so we know that the hypotenuse has length root 58, the side opposite A has length 3, and the side adjacent to A has length 7. And since we know our definition of sine, cosine, and tangent, we can use those to find the sine of A, the cosine of A, and the tangent of A. So the sine of A is the length of the opposite side over the length of the hypotenuse. So we'll substitute in those values and get... The cosine of A is the adjacent length over the hypotenuse, so we'll substitute in those values. And the tangent is the opposite over the adjacent, so we'll substitute in those values. It's important to remember that the terms adjacent and opposite depend on where the angle is located. So if I were to take a look at the sine, cosine, and tangent of B, my opposite side and my adjacent side are different. The hypotenuse is still the side across from the right angle, that's still square root of 58. But now if I take a look at the side opposite B, that has length 7. And the length of the side adjacent to B has length 3. And it's still true that sine is opposite over hypotenuse, cosine is still adjacent over hypotenuse, and tangent is still opposite over adjacent. So I can substitute in the lengths that we know for these sides and find the sine, cosine, and tangent. Or we might have a triangle that looks like this, and we want to find the sine, cosine, and tangent of our angle, which means we need to find the length of the third side of the triangle. So again, the hypotenuse C is always the side across from the right angle, this side. Little A is the side across from angle A, and little B is the side across from angle B. And we can use our Pythagorean theorem to find the length of the third side. And that tells us that the hypotenuse is 13, the side adjacent to A has length 5, and the side opposite of A has length 12. So we can find the sine, cosine, and tangent.