Question

How do you evaluate `sin(cos^-1(2/3))` without a calculator?

Answer 1

`sqrt 5/3`

Explanation:

Let `a = cos^(2/3) in Q1`, Then, `cos a = 2/3`.

The given expression is

`sin a = +-sqrt(1-cos^2 a) = +-sqrt(1-(2/3)^2) = +-sqrt 5/3`

As a is in Q1, negative sign is inadmissible.

For -a also, the cosine is 2/3 but the sine is `-sqrt 5/3`.

Calculators use programs for the principal value only. ..