The trig functions & right triangle trig ratios

on the right hand side we have a bunch of expressions that are just ratios of different information given in these two diagrams and then over here on the Left we have the sign taken of angle mkj cosine of angle mkj and tangent of angle mkj and angle mkj is this angle right over here same thing as theta so these two angles these two angles have the same measure we see that right over there and what we want to do is figure out which of these expressions are equivalent to which of these expressions right over here and so I encourage you to pause the video and try to work this through on your own so assuming you've had a go at it so let's try to work this out and when you look at this diagram it looks like the intention here on the left is this evokes the unit circle definition of trig functions because this is really a this is a this is a unit circle right over here and this evokes kind of the sohcahtoa definition because it's we're just kind of in a plain vanilla right triangle and so just to remind ourselves let's just remind ourselves of sohcahtoa because have a feeling it might be useful so sine is opposite over hypotenuse cosine is adjacent over hypotenuse over hypotenuse and tangent is opposite over adjacent so we can refer to this and we can also refer to remind ourselves of the unit circle definition of trig functions that the cosine of an angle is the x-coordinate and that the sine of where this this of where this ray intersects the unit circle and the sine of this angle is going to be the y-coordinate and what we'll see through this video is that they're they're actually the unit circle definition it's just an extension of sohcahtoa so let's look first at x over 1 so we have X X as the x-coordinate that's also the length of this side right over here relative to this angle theta that is the adjacent side so X is equal to the adjacent side what is 1 well this is a unit circle 1 is the length of the radius which for this right triangle is also the hypotenuse so if we apply the sohcahtoa definition x over 1 is adjacent over hypotenuse adjacent over hypotenuse adjacent over hypotenuse that's cosine so that's going to be this is equal to cosine of theta but theta is the same thing as angle mkj they have the same measure so cosine of angle mkj is equal to cosine of theta which is equal to x over 1 now let's move over to Y over 1 well Y is going to be the length of this side right over here Y is going to be Y let me do this in the blue Y is going to be this length relative to angle theta that is the opposite side that is the opposite side now which trig function is opposite over hypotenuse opposite over hypotenuse that's sine of theta sine of theta so sine of angle mkj is the same thing as sine of theta we see that they have the same measure and now we see that's the same thing as Y over 1 now for both of these I use the sohcahtoa definition but we could have also used the unit circle definition x over 1 that's the same thing that's the same thing as X and the unit circle definition says well this X this point the x-coordinate of where this I guess you could say the terminal side of this angle this ray right over your intersects the unit circle that by definition by the unit circle definition is the cosine of this angle it X is equal to the cosine of this angle and the unit circle definition the y-coordinate is equal to the sine of this angle we could have written this we could have written this as instead of X comma Y we could have written this as cosine of theta sine theta just like that but let's keep going now we have x over Y we have adjacent over we have adjacent over opposite so this is equal to adjacent over opposite tangent is opposite over adjacent not adjacent over opposite so this is the reciprocal of tangent so this right over here if we had to this is equal to 1 over tangent of theta and we later learn about cotangent and all of that which is essentially this but it's not one of our so we can rule this one out but then we have y over X well this this is looking good this is y is opposite opposite X is adjacent relative to angle theta adjacent so this is the tangent of theta this is equal to tangent of theta so tangent of angle mkj is the same thing as tangent of theta which is equal which is equal to Y over X now let's look at J over K so J over K now we're moving over to this triangle J over K so relative to this angle because this is the angle that we care about J is the length of the adjacent side and K is the length of the opposite side of the opposite side so this is adjacent over opposite so this is equal to adjacent over opposite tangent is opposite over adjacent not adjacent over opposite so once again this is 1 this is the reciprocal of the tangent function not one of the choices right over here so we can rule that one out now K over J well now this is opposite over adjacent opposite over adjacent that is equal to tangent of theta this is equal to tangent of theta or tangent of angle mkj so this is equal to K over J now we have M over J M over J hypotenuse over adjacent side this of course this of course is equal to the hypotenuse hypotenuse over adjacent well if it was adjacent over hypotenuse we'd be dealing with cosine but this is the reciprocal of that so this is actually 1 over the cosine of theta not one of our choices not one of our choices here so I'll just rule that one out right over there but then we have its reciprocal reciprocal J over m that's adjacent over hypotenuse adjacent over hypotenuse is cosine so this is equal to cosine of theta or cosine sine of angle mkj so we could write it down so this is equivalent to J over m and then one last one K over m well that's opposite over hypotenuse opposite over hypotenuse that's going to be sine of theta so this right over here is equal to sine of theta which is the same thing as sine of angle mkj which is the same thing as all of these expressions so this is equal to K over m and we are done