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Sal finds all six trigonometric ratios (sine, cosine, tangent, secant, cosecant, and cotangent) of an angle in a given right triangle. Created by Sal Khan.
Video transcript
Determine the six trigonometric ratios for angle A in the right triangle below. So this right here is angle A, its in vertex A and help me remember the definitions of the trig ratios, these are human constructed definitions that have ended up being very, very useful for analyzing a whole series of things in the world and to help me remember them, I use the word, SOH-CAH-TOA let me write that down, so SOH-CAH-TOA sometimes you can think of it as one word, but its really the three parts that define atleast three of the trig functions for you, then we can get the other three by looking at the first three, so SOH tells us that sin of an angle, in this case its sin of A, so sin of A is equal to the opposite opposite, thats the o over the hypotenuse so opposite over the hypotenuse, well in this context what is the opposite sin to angle A? so we go across the triangle it opens up onto side BC, it has length 12, so that is the opposite side. So this is going to be equal to 12, and whats the hypotenuse? well, the hypotenuse is the longest side of the triangle; it's opposite to the 90 degree angle, and so we go opposite the 90 degree angle the longest side is side AB, it has length 13, so this side right over here is the hypotenuse, and so the sin of A is 12/13ths. now lets go to CAH CAH defines, cosine for us. it tells us that cosine of an angle, in this case, cosine of A is equal to the adjacent side, the adjacent side to the angle over the hypotenuse, over the hypotenuse. so whats the adjacent side to angle A? well if we look at angle A, there is 2 sides next to it. One of them is the hypotenuse the other one has length 5, the adjacent one is side CA so its 5, and what is the hypotenuse, well we've already figured that out, the hypotenuse right over here its opposite the 90 degree angle, it the longest side of the right triangle, it has length 13, so the cosine of A is 5/13ths and let me label this, this right over here is the adjacent side and this is all specific to angle A, the hypotenuse would be the same regardless of what angle you pick, but the opposite and adjacent is dependent on the angle we choose in the right triangle, now lets go to TOA. TOA defines tangent for us, it tells us that the tangent the tangent of an angle is equal to the opposite, equal to the opposite side over the adjacent side, so given this definition what is the tangent of A? well the opposite we already figured out has length 12, and the adjacent side we already figured out has length 5 so the tangent of A, which is opposite over adjacent is 12/5ths now we'll go to the other three trig ratios which you can think of as the reciprocals of these right over here, but I'll define them so first you have cosecant, and cosecant; its always a little bit unintuitive, why cosecant is the reciprocal of sine of A even though it starts with a co like cosine but cosecant is the reciprocal of the sin of A, so sin of A is opposite over hypotenuse. Cosecant of A is hypotenuse over opposite and so whats the hypotenuse over the opposite, well hypotenuse is 13 and the opposite side is 12 and notice 13/12ths is the reciprocal of 12/13ths now, secant of A is the reciprocal so instead of it being adjacent over hypotenuse, which we got out of the CAH part of SOH CAH TOA, its hypotenuse over adjacent so what is the secant of A? well the hypotenuse, we figured out multiple times is 13 and what is the adjacent side? it is 5, so 13 13/5ths, which is once again the reciprocal of cosine of A, 5/13ths finally lets get the cotangent, and the cotangent is the reciprocal of the tangent of A, instead of being opposite over adjacent it is adjacent over opposite, so what is the cotangent of A? well we figured out the adjacent side multiple times for angle A, its 5 and the opposite side to angle A is 12, so 5/12ths which is once again the reciprocal of the tangent of A which is 12/5ths