How Do Cosine and Sine Relate to Arctan in Trigonometry?

[rainyrabbit]

Why is cos(arctan x) = 1 / root(1+x^2)

and sin(arctan x) = x / root(1+x^2)?

I would greatly appreciate help. I'm studying for physics and this came up in one of the solutions. Thank you.

 

[dodo]

Think of a right triangle with sides 1 and x. Now try to imagine the values of:
- The hypotenuse, and
- The sin, cos and tan of the angle next to the "1" side.

 

[arildno]

Well, you could show this algebraically, however:

What is arctan(x)? arctan(x) is that angle whose tan-value is x.

Now, writing x=x/1, remember that tan can be thought of as the ratio between two of the sides in a right-angled triangle.

Identify which sides these are, and see if you can deduce the results for yourself.