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

Aug 7, 2016

$\frac{\sqrt{5}}{3}$

#### Explanation:

Let $a = {\cos}^{\frac{2}{3}} \in Q 1$, Then, $\cos a = \frac{2}{3}$.

The given expression is

$\sin a = \pm \sqrt{1 - {\cos}^{2} a} = \pm \sqrt{1 - {\left(\frac{2}{3}\right)}^{2}} = \pm \frac{\sqrt{5}}{3}$

As a is in Q1, negative sign is inadmissible.

For -a also, the cosine is 2/3 but the sine is $- \frac{\sqrt{5}}{3}$.

Calculators use programs for the principal value only. ..