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determine the six trigonometric ratios for angle a in the right triangle below so this right over here is angle a it's at vertex a and to help me remember the definitions of the trig ratios and these are human constructed definitions that have ended up being very very useful for analyzing a whole series of things in the world but to help me remember them I use the word sohcahtoa so let me write that down so so so CUH so Toa sometimes you can think oh there's one word but it's really the three parts that define at least three of the trig functions for you and then we can get the other three by looking at the first three so Soh tells us that sine of an angle in this case it's sine of a so sine of a is equal to the opposite opposite that's the O over the hypotenuse opposite over the hypotenuse well in this context what is the opposite side to angle a well 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 what's the hypotenuse well the hypotenuse is the longest side of the triangle it's opposite the 90-degree angle and so we go opposite the 90-degree angle longest side aside a B it has length 13 so this right over here is the hypotenuse so the sine of a is 12 13 now let's go to cut cod defines cosign 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 what's the adjacent side to angle a well if we look at angle a there's two sides that are next to it one of them is the hypotenuse the other one has length five the adjacent one is side C a so it's five and what is the hypotenuse well we've already figured that out the hypotenuse right over here it's opposite the 90 angle it's the longest side of the right triangle it has length 13 so the cosine of a is 513 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 the adjacent is dependent on the angle that we choose in the right triangle now let's go to Toa Toa defines tangent force 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 side we already figured out has length 12 as 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 fifths now we'll go to the other three trig ratios which you could think of as the reciprocals of these right over here but I'll define it so first you have cosecant and cosecant it's always a little bit unintuitive why cosecant is the reciprocal of sine of a even though it starts with a KO like cosine but cosecant deserves typical of the sine of a so sine of a is opposite over hypotenuse cosecant of a is hypotenuse over opposite and so what's the hypotenuse over the opposite well the hypotenuse is 13 and the opposite side is 12 I notice 13 12 is a reciprocal of 12 13 now secant of a it's the reciprocal so instead of being adjacent over hypotenuse which we got from the cut part of sohcahtoa its hypotenuse its hypotenuse over adjacent so what is the secant of a well the hypotenuse we've figured out multiple times already is 13 and what is the adjacent side it's five so it's 13 fifths which is once again the reciprocal of the cosine of a 513 finally let's get the cotangent and the cotangent is the reciprocal of tangent of a instead of being opposite over adjacent it is adjacent over opposite so what is the code tangent of a well we figured out the adjacent side multiple times for angle eighths length five and the opposite side to angle a is 12 so it's five twelfths which is once again the reciprocal of the tangent of a which is 12 fifths