 So sine, cosine, and tangent really are the only trigonometric ratios you need to know, but it's occasionally useful to deal with three other trigonometric ratios. So again, we'll define them in terms of a right triangle and an acute angle with measure A. The trigonometric ratio secant is the length of hypotenuse over the adjacent side, cosecant is hypotenuse over opposite, and cotangent is adjacent over opposite. Now it's worth comparing the definitions of these three extra trigonometric ratios to the original sine, cosine, and tangent. If we do that, notice that secant is the reciprocal of cosine. Secant is length of the hypotenuse over the length of the adjacent, while cosine is the length of the adjacent over the length of the hypotenuse. Similarly, cosecant is the reciprocal of sine. Cosecant is the length of the hypotenuse over the length of the opposite side, and sine is the length of the opposite side over the length of the hypotenuse. And not to be left out, cotangent is the reciprocal of tangent, cotangent is the length of the adjacent over the opposite, while tangent is the length of the opposite over the adjacent. And so it's actually more convenient to remember these other three trigonometric ratios as follows. Secant is one over cosine, cosecant is one over sine, and cotangent is one over tangent. And a helpful thing to remember is that secant, the s-function, is the reciprocal of cosine, the c-function, and cosecant, the c-function, is the reciprocal of sine, the s-function. C and s always go together, except for cotangent and tangent. There, you just have to remember that cotangent probably has something to do with tangent, and since everything else is defined in terms of a reciprocal, it's going to be the reciprocal of tangent. So let's find the values of these six trigonometric ratios for angle A. Since we need all three sides of the right triangle and only have two, we use the Pythagorean theorem to find the third side. And we know the hypotenuse is 8 and one side has length 3, so we'll substitute, then solve, where, because B is supposed to be a length, we'll take the positive square root. Remember, definitions are the whole of mathematics, all else is commentary, and so we'll pull in our definitions for sine, cosine, and tangent. And we see we need to identify the length of the opposite side, the adjacent side, and the hypotenuse. So we find these three sides. Sine is opposite over hypotenuse. Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. Once we have the values for sine, cosine, and tangent, it's easy to find the values for secant, cosecant, and cotangent because they are just the reciprocals. So we'll pull in those definitions from the reciprocals. So secant is the reciprocal of cosine. Cosecant is the reciprocal of sine. And cotangent is the reciprocal of tangent.