Question

How would I find the cos(tan^-1(x)) without the use of a calculator? My teacher did a very poor job explaining...

Answer 1

Well, assuming you have an x. Like for example, 1. The question would basically be "given a triangle that has a tangent valueof 1, what is the cosine of that triangle?" So that being said, let's say we have a cos(tan^-1(1)) I'll draw a triangle:



Now recall that tangent in reference to a triangle means opposite/adjacent. And the only way for a tangent value to be 1 is if the value of the opposite and adjacent sides are the same. So now keep in mind what we want to do "given a triangle with a tangent value of 1, what is the cosine of that triangle?" So cosine means adjacent over hypotenuse. So we need to find the hypotenuse. This is done with regular pythagorean theorem of
\[c=\sqrt{a ^{2}+b ^{2}} \]So if a and b are both 1, I get this for C
\[c=\sqrt{1+1}=\sqrt{2} \]So now that I have the hypotenuse, I can say cosine is adjacent over hypotenuse. Meaning my cosine value is
\[\frac{ 1 }{ \sqrt{2} }=\frac{ \sqrt{2} }{ 2 } \]
And this would be my final answer. Kinda make sense?

Answer 2

Thank you so very much! That is a much better explanation, I really appreciate it. I wish my professor who I'm paying money could have done better, but I guess I'm just going to have to rely on my own studies to get by. Thanks once again!

Answer 3

Yeah, np at all ^_^

Answer 4

let\[\tan^{-1} x=\theta ,x=\tan \theta \]

Answer 5

\[\cos \theta=\frac{ 1 }{\sqrt{1+x ^{2}} }\]