Question

How do you simplify `sin(2cos^-1(4/5))`?

Answer 1

Use the identity `sin(2A)=2sin(A)cos(A)`
Then use the identity `sin(A) = +-sqrt(1-cos^2(A))`

Explanation:

Given: `sin(2cos^-1(4/5))`

Use the identity `sin(2A)=2sin(A)cos(A)`

`2sin(cos^-1(4/5))cos(cos^-1(4/5))`

The cosine of its inverse yields its argument:

`2sin(cos^-1(4/5))(4/5)`

Perform the multiplication:

`8/5sin(cos^-1(4/5))`

Use the identity #sin(A) = +-sqrt(1-cos^2(A))

`+-8/5sqrt(1-cos^2(cos^-1(4/5)))`

Again, the cosine of its inverse yields its argument:

`+-8/5sqrt(1-(4/5)^2)`

`+-8/5sqrt(25/25-16/25)`

`sin(2cos^-1(4/5)) = +-24/25`

To determine whether to choose the positive or negative value, one would need know whether the angle was in the first or fourth quadrant.