>> We're given cosecant X equals 4 for some angle between 90 and 180 degrees,
which means it terminates in the second quadrant.
We want to find sin of X divided by 2, cosine X divided by 2, and tension X divided by 2.
Well if we know cosecant X equals 4 or if we want 4 over 1, since cosecant is a reciprocal
of sin, this would give us the ratio of the hypotenuse
to the opposite side of our reference triangle.
So let's sketch X in center position, which we know terminates in the second quadrant,
so it might look something like this.
So this would be X degrees.
And now if we sketch the reference triangle for this angle, it would be this triangle here.
So this would be our reference angle, and the ratio of the hypotenuse
to the opposite side would be 4 to 1, so we can label the hypotenuse 4, the opposite side 1,
and now we can use the Pythagorean theorem to find this other leg,
which would be the square root of 4 squared minus 1 squared, or the square root of 15.
So we'll label this as square root of 15, but because we're in the second quadrant,
where X is negative, this would be negative as well.
And now we'll use one of our half angle identities
to determine the value of sin X over 2.
One thing to notice, though, is if we take an angle between 90 and 180 degrees and divide it
by 2, notice how X over 2 would be greater than 45 degrees and less than 90 degrees.
Which means the angle X divided by 2 would be in the first quadrant,
where we know all of these trick function values would be positive.
So we don't have to worry about the plus or minus sign here,
we know all of our trick function values here will be positive.
Now using the half angle identity for sin A divided by 2, or in our case,
sin X divided by 2, we would have -- we know it's going to be positive.
We'd have 1 minus cosine, in our case, X divided by 2.
Now we can find the value of cosine X using our reference triangle, so we'd have the square root
of 1 minus cosine X is equal to the ratio of the adjacent of the hypotenuse,
which would be negative square root of 15 divided by 4.
All this is divided by 2.
And now we can go to the calculator to get an approximation for sin X divided by 2.
We would have the square root and then we have a numerator and a denominator,
so we'll put the numerator in parentheses, open parenthesis 1.
Notice how this would become plus because we have minus and negative.
So we have plus square root 15, right arrow, divided by 4,
close parenthesis, and then we'll divide this by 2.
You'll have one more parenthesis to end the larger square root.
And enter.
So the sin function value is approximately 0.9920.
And now we can easily find the cosine function value or cosine of X divided by 2.
This would be equal to the square root of 1 plus cosine X divided by 2.
So we just have a difference in the sign here.
We'd have the square root of 1 plus negative square root 15 divided by 4, all over 2.
And now we'll go back to the calculator.
And because we only have a sign change here, let's go ahead and press second enter
which brings up the previous entry, and now we can just change this plus sign here to minus.
1 minus this fraction is equivalent to 1 plus the negative fraction.
So now we'll press enter and the cosine function value of X over 2 would be approximately 0.1260.
Now to find the tangent function value we could use one of these identities here,
but since tangent theta is equal to the [inaudible] divided by cosine data,
let's go ahead and use the fact that tangent of X divided by 2 is equal to sin of X divided
by 2 divided by cosine of X divided by 2.
And we can find this quickly on the calculator.
We'd have .9920 divided by .1260,
so the approximately tangent function value of X over 2 would be 7.8730.
I hope you found this helpful. ------------------------------15f91d32823f--