Question

Exact value of cos45

Answer 1

It's a value you should probably memorize:

$\cos45^\circ=\dfrac{\sqrt2}2=\dfrac1{\sqrt2}$

You can derive it using some trigonometric identities, other known values of cosine, and properties of the cosine function. For example, using the double angle identity for cosine:

$\cos^2x=\dfrac{1+\cos2x}2$

If $x=45^\circ$, then

$\cos^245^\circ=\dfrac{1+\cos90^\circ}2$

and you probably know that $\cos90^\circ=0$, so

$\cos^245^\circ=\dfrac12$

When we take the square root, we should take the positive root because $\cos x>0$ whenever $0^\circ$:

$\cos45^\circ=+\sqrt{\dfrac12}\implies\cos45^\circ=\dfrac1{\sqrt2}$