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

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Answer 1 · 2017-11-15T18:06:00

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

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.

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