Using the cosine double-angle identity

we have triangle ABC here which looks like a right triangle we know it's a right triangle because 3 squared plus 4 squared is equal to 5 squared and they want us to figure out what cosine of 2 times angle ABC is so that's this angle a b c well we can't immediately evaluate that but we do know what what the cosine of angle ABC is we know that the cosine of angle a b c well cosine is just adjacent over hypotenuse is going to be equal to 3/5 and similarly similarly we know what the sine of angle a b c is that's opposite over hypotenuse that is 4/5 so if we could break this down into just cosines of ABC and sines of ABC then we'll be able to evaluate it and lucky for us we have a trig identity at our disposal that does exactly that we know we know that the cosine of 2 times 2 times an angle is equal to cosine of that angle squared minus sine of that angle squared we prove this in other videos but this becomes very helpful for us here because now we know that the cosine I'm going to do this in a different color now we know that the cosine of angle a B C is going to be equal to a sorry the cosine of 2 times the angle ABC that's what we care about 2 times the angle ABC is going to be equal to the cosine of angle ABC squared squared minus sine of the angle a B C squared a B C squared and we know what these things are this thing right over here this thing right over here is just going to be equal to 3/5 squared cosine of angle ABC is 3/5 so we're going to square it and this right over here is just 4 v squared so it's minus 4/5 squared and so this simplifies to 9 over 25 minus minus 16 over 20 five which is equal to which is equal to seven over 25 so this thing right over here is equal to seven sorry it's negative got to be careful there 16 is larger than 9 negative negative 7 over 25 now one thing that might jump at you is why why did I get a negative value here when when I when I doubled the angle here because the cosine was clearly a positive number and there you just have to think of the unit circle which we already know is an extension is an extension the unit circle definition of trig functions which is an extension of the sohcahtoa definition x-axis y-axis let me draw a unit circle here my best attempt so that's our unit circle so this angle right over here looks like something like this so it looks like something like this and so you see its x-coordinate which is the cosine of that angle looks positive but then if you were to double this angle it would take you out someplace like this it would take you out someplace like this and then you see by the unit circle definition the x-coordinate is now we are now sitting in the second quadrant and the x-coordinate can be negative and that's essentially what happened in this problem