Question

Why is abs(cos theta) never greater than 1 in the first place? (this is linear algebra)

Answer 1

actually it looks like trig
cosine is the first coordinate on the unit circle (at least that is one way to think of it) so it is never bigger than 1 or smaller than -1

Answer 2

But I need mathematical proofs to show why it's never greater than 1.

Answer 3

i guess that depends entirely on your definition of cosine

Answer 4

Well, this is like the first couple chapters of linear algebra.

Answer 5

I know that cos 0=1.

Answer 6

So the domain is [-1, 1] for cos theta, right?

Answer 7

@electrokid

Answer 8

the "range" for sine and cosine functions is \([-1,1]\)

Answer 9

you can prove from the identity
\[
\begin{align*}
\sin^2x+\cos^2x&=1\\
\implies\cos^2x&=1-\sin^2x
\end{align*}
\]
since the square of a number is always positive, the right hand side is always less than "1" but must be >0
hence the square-root must also be in that range.
you can prove similarly arguing using Pythagorean thm.