Find the exact value of sin (arccos (three fifths)). for full credit, explain your reasoning.

We want to evaluate sin (cos⁻¹ (3/5)).

Note that

sin²x + cos²x = 1

Therefore

sin²x = 1 - cos²x

Let y = cos⁻¹ (3/5)

Then

cos (y) = 3/5

sin²y = 1 - cos²y = 1 - 9/25 = 16/25

sin (y) = 4/5

Therefore

sin (cos⁻¹ (3/5)) = sin (y) = 4/5

Answer: 4/5