Question

How do you evaluate `sec(cos^-1(1/3))` without a calculator?

Answer 1

`sec(cos^-1(1/3))=3`

The restriction for the range of arccos x is `[0,pi]`. Since the argument is positive it means our triangle is in quadrant I with adjacent of 1 and hypotenuse 3. Hence using pythagorean theorem we find that the opposite is `2sqrt2`.

Now let find the ratio for the secant of the angle from the triangle which is `sec theta=h/a=3/1=3`