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

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Answer 1 · 2016-08-07T11:46:21

$sqrt 5/3$

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. ..

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