Question

Exact value of cos45

Answer 1

It's a value you should probably memorize:

`\cos45^\circ=\frac{\sqrt2}2=\frac1{\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=\frac{1+\cos2x}2`

If
`x=45^\circ`, then

`\cos^245^\circ=\frac{1+\cos90^\circ}2`

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

`\cos^245^\circ=\frac12`

When we take the square root, we should take the positive root because
`\cos x>0` whenever
`0^\circ<x<90^\circ`:

`\cos45^\circ=+√(\frac12)\implies\cos45^\circ=\frac1{\sqrt2}`