Question

How do you simplify `cos^-1(sqrt2/2)`?

Answer 1

`cos^-1(sqrt2/2)=pi/4`

Explanation:

This value is actually a standard value in disguise. Surds are often removed from the denominator of a fraction by multiplying top and bottom by the surd in question. We can do exactly the same thing to put the surd back in the denominator:

`sqrt2/2*sqrt2/sqrt2=2/(2sqrt2)=1/sqrt2`

Maybe you recognise the value now.

What value of theta gives this value? It is one that should be memorised if you have exams on trigonometry.

`cos^-1(sqrt2/2)=cos^-1(1/sqrt2)=pi/4`

Writing this another way, we have:

`cos(theta)=sqrt2/2=1/sqrt2`

`theta=pi/4`

Answer 2

`pi/4; (7pi)/4`

Explanation:

Trig table and unit circle give -->
`cos x = sqrt2/2` --> arc `x = +- pi/4`,
or, using co-terminal arcs:
`x = pi/4` and `x = (7pi)/4`