We want to find the exact value
of the expression.
We are given cosine of two times inverse sine of -12/13.
The first thing to recognize here is that the inverse sine
of -12/13 is equal to an angle theta,
that has a sine function value equal
to the input into the inverse sine function,
and the angle theta is in the closed interval
from negative 90 degrees to positive 90 degrees.
So in our case, the angle theta
has a sine function value of -12/13.
So if we let inverse sine of -12/13 equal theta,
we can say the given expression is equal to cosine 2 theta.
The next step, we will model the angle theta
on the coordinate plane,
and then sketch the reference triangle.
Notice here, because the sine function value is negative,
we know the angle theta must be negative
and terminate in the fourth quadrant.
So let's say the angle theta terminates here,
and therefore here's the angle theta.
And now let's sketch the reference triangle.
Because the sine function value is equal to -12/13,
this gives us the ratio of the opposite side to hypotenuses
for the reference triangle.
And we know the hypotenuses is always positive,
and therefore we label the opposite side -12
and the hypotenuses 13.
And this should make sense
because y is negative in the fourth quadrant.
And now let's find the length of the adjacent side
using the Pythagorean theorem.
We may recognize this is a 5, 12, 13, right triangle,
and therefore the adjacent side has the length of 5 units,
but let's verify this using the Pythagorean theorem.
And let's use the Pythagorean theorem
in the form of a squared plus b squared equals c squared.
Let's let a equal the unknown length,
which gives us a squared plus, let's let b equal -12.
So we have, plus the square of -12 equals c squared,
where c squared is the square of 13.
Simplifying, we have a squared plus 144 equals 169.
To isolate a squared we subtract 144 on both sides,
which gives us a squared equals 25.
And now to solve for a
we take the square root of both sides of the equation.
Algebraically, we have two solutions,
one positive and one negative.
So we have a equals plus or minus 5,
but because we're in the fourth quadrant
where x is positive, we know a has to be positive 5.
Now that we have the reference triangle for the angle theta,
we can use a double angle identity
and evaluate cosine 2 theta.
Notice there are three options for cosine 2a.
Let's go ahead and just use the first one,
since cosine 2a is equal to cosine squared a
minus sine squared a.
We have cosine 2 theta is equal to 2.
We have cosine 2 theta is equal to cosine squared theta
minus sine squared theta.
And now we can determine
the cosine and sine function values,
using the reference triangle.
Cosine theta is equal to the ratio
of the adjacent side to the hypotenuse, which gives us 5/13.
So we have the square of 5/13 minus,
sine theta is equal to the ratio
of the opposite side to the hypotenuse, which is -12/13,
which was also given in the original expression,
giving us the square of -12/13.
And now simplifying.
To square a fraction,
we square the numerator and denominator.
The square of 5/13 is equal to 25/169
minus the square of -12/13 is 144/169.
And now subtracting, the denominator stays the same,
and we subtract the numerators.
25 minus 144 is equal to -119,
and therefore the given expression is equal to -119/169.
I hope you found this helpful.