Hansard"s parliamentary debates

     
Determine the sixtrigonometric ratios for angle A in theright triangle below. So this right over here isangle A, it's at vertex A. Và to help me rememberthe definitions of the trig ratios-- & these are humanconstructed definitions that have ended up being very,very useful for analyzing a whole series ofthings in the world. To help me remember them, Iuse the words soh cah toa. Let me write that down. Soh cah toa. Sometimes you can thinkof it as one word, but it's really the three partsthat define at least three of the trig functions for you. Và then we canget the other three by looking at the first three. So soh tells us that sine of anangle-- in this case it's sine of A-- so sine of A is equalto the opposite, that's the O, over the hypotenuse. Well in this context, what isthe 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 goingto be equal to 12. And what's the hypotenuse? Well, the hypotenuse is thelongest side of the triangle. It's opposite the90 degree angle. Và so we go oppositethe 90 degree angle, longest side is side AB. It has length 13. So this right overhere is the hypotenuse. So, the sine of A is 12/13. Now let's go to cah. Cah defines cosine for us. It tells us that cosine ofan angle-- in this case, cosine of A-- is equalto the adjacent side khổng lồ the angle overthe hypotenuse. So, what's the adjacentside lớn angle A? Well, if we lookat angle A, there are two sides thatare next lớn it. One of them is the hypotenuse. The other one has length 5. The adjacent one is side CA. So it's 5. & what is the hypotenuse? Well, we've alreadyfigure that out. The hypotenuse isright over here, it's opposite the90 degree angle. It's the longest sideof the right triangle. It has length 13.


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So the cosine of A is 5/13. And let me label this. This right over hereis the adjacent side. Và this is allspecific khổng lồ angle A. The hypotenuse would be thesame regardless of what angle you pick, but theopposite & the adjacent is dependent onthe angle that we choose in the right triangle. Now let's go to toa. Toa defines tangent for us. It tells us that thetangent of an angle is equal to lớn the oppositeside 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. Và the adjacent side,we already figure out, has length 5. So the tangent of A, whichis opposite over adjacent, is 12/5. Now, we'll go the to lớn theother three trig ratios, which you could think ofas the reciprocals of these right over here. But I'll define it. So first you have cosecant. & cosecant, it's alwaysa little bit unintuitive why cosecant is thereciprocal of sine of A, even though it startswith a co like cosine. But, cosecant is thereciprocal of the sine of A. So sine of A is oppositeover hypotenuse. Cosecant of A ishypotenuse over opposite. Và so what's the hypotenuseover the opposite? Well, the hypotenuse is 13and the opposite side is 12. And notice that 13/12 isthe reciprocal of 12/13. Now, secant of Ais the reciprocal. So instead of beingadjacent over hypotenuse, which we got from thecah part of soh cah toa, it's hypotenuse over adjacent. So what is the secant of A? Well, the hypotenuse, we'vefigured out multiple times already, is 13. Và what is the adjacent side? It's 5. So it's 13/5, whichis, once again, the reciprocal of thecosine of A, 5/13. Finally, let'sget the cotangent. Và the cotangent is thereciprocal of tangent of A. Instead of beingopposite over adjacent, it is adjacent over opposite. So what is the cotangent of A? Well, we've figuredout the adjacent side multiple times forangle A. It's length 5. Và the opposite sideto angle A is 12. So it's 5/12, which is,once again, the reciprocal of the tangent ofA, which is 12/5.