Find the cosine of u divided by two
if cosine u equals negative 2.381
and angle u is in quadrant four.
We'll determine cosine of u divided by two
using the half-angle identity cosine
of a divided by two equals plus or minus the square root
of the quantity one plus cosine a divided by two.
Notice how to find cosine of u divided by two, we will have
to find the value of cosine u and therefore let's sketch
the reference triangle for angle u in quadrant four.
Let's assume this is the terminal side of angle u
in quadrant four.
And therefore this acute angle here is the reference angle.
Let's call it angle u sub r.
And this would be the reference triangle.
Because we are in quadrant four, x is positive
and y is negative.
We are also given cosecant u equals negative 2.381.
Cosecant u using right triangle trigonometry is equal to
the ratio of the hypotenuse to the opposite side
or on the coordinate plane, r divided by y.
And because we are in quadrant four where x is positive
and y is negative and r is always positive,
let's write the cosecant functional value as cosecant u
equals, looks like this is a fraction where y is negative
and therefore they'll have positive 2.381 over negative one.
Notice how this quotient is still negative 2.381
but r is positive and y is negative which is true
in quadrant four.
So going back to the reference triangle, we will label
the hypotenuse 2.381 in the opposite side
of the reference triangle negative one.
So the hypotenuse or r is positive 2.381
and the opposite side of y is negative one.
And this leg here would be x.
And because we need cosine u, we will have to find x
or the length of this leg here which will be positive
because we are in quadrant four.
So plugging in the Pythagorean Theorem we'll export this
y-squared equals r-squared.
We have x-squared plus the square root of negative one
is equal to the square of 2.381.
Let's isolate x-squared by subtracting the square root
of one on both sides, which gives us x-squared is equal
to the square of positive 2.381
minus the square of negative one.
Now, we take the square root of both sides of the equation.
And then we have a plus or minus on the right,
but again we already know x is positive because we are
in quadrant four.
Simplifying, we have x is equal to this square root
which we will approximate on the calculator.
We have the square root of the square of 2.381
minus the square of negative one
which is approximately 2.160824.
Let's go ahead and label this on the reference triangle.
And now we have all the information we need in order to find
cosine of u divided by two.
Cosine of u divided by two is equal to plus or minus
the square root with the quantity one plus cosine u
all divided by two.
We now need to determine the sine
of cosine of u divided by two.
Because the angle u is in quadrant four, we know u is
between 270 and 360 degrees.
If we divide everything by two, the angle u divided by two
is between 135 and 180 degrees which is in quadrant two.
In quadrant two, x is negative and y is positive
because x is negative, cosine is negative and therefore
we have negative square root of the quantity
one plus cosine u.
Going back to the reference triangle, cosine u is equal to
x divided by r or the ratio of the adjacent side
to the hypotenuse,
which is 2.160824 divided by 2.381.
And all this is still divided by two under the square root.
And now I go back to the calculator
to get a decimal approximation
for cosine of u divided by two.
We have negative and then the square root.
We need this numeral in parentheses, so we have open
parenthesis, one plus new parentheses, 2.160824
divided by 2.381,
close parenthesis for this quotient,
close parenthesis for the new enter and then divided by two.
You're still under the square root.
And then enter.
Cosine of u divided by two down to four decimal places
is approximately negative 0.9766.
I hope you found this helpful.