Question

How do you find the exact value of `cos^-1 (sqrt2/2)`?

Answer 1

See below

Explanation:

Let `theta = cos^-1(sqrt2/2)`

`costheta=sqrt2/2`

So in an imaginary right-angled triangle, the length of the `"hyp"` is `2` and the length of the `"adj"` is `sqrt2`. This means that the length of the `"opp"` is `sqrt(2^2-(sqrt2)^2)=sqrt2`.

Since the `"adj"` and `"opp"` are equal lengths, our triangle is isosceles. This means that it also has two angles of equal lengths. Since one angle is `pi/2`, the other two must be `pi/4`.

So if we say that `theta=pi/4`, then `cos(pi/4)="adj"/"hyp"=sqrt2/2`
`thereforepi/4=cos^-1(sqrt2/2)`