We're asked to find sine of u divided by two
if sine u equals negative 0.632
and angle u is in quadrant four.
We will determine sine of u divided by two
using the half-angle identity sine of A divided by two
equals plus or minus the square root
of the quantity one minus cosine A divided by two.
Notice to find sine of u divided by two,
we will have to find cosine of angle u.
To do this, let's use a reference triangle
on the coordinate plane.
Let's first sketch the terminal side of angle u
in quadrant four.
Let's assume this is the terminal side of angle u.
And therefore, this acute angle here
is the reference angle.
We'll call it angle u sub r.
And now we will sketch the reference triangle,
which should be this triangle here.
Because we are in quadrant four,
notice x is positive and y is negative.
And using right triangle trigonometry,
we know that sine u
is equal to the ratio of the opposite side
to the hypotenuse,
or in the coordinate plane,
sine u is equal to y divided by r,
where r is always positive
and because we are in quadrant four,
y is negative.
And therefore, let's write the sine function value
as a fraction
as negative 0.632, because y is negative,
divided by one, because you know r is positive.
Notice how we still have a sine function value
of negative 0.632,
but now we can label y or the opposite side
of the reference triangle, negative 0.632
and r the hypotenuse, positive one.
So let's go ahead and label that.
We have negative 0.632 for y or the opposite side
and one for r or the hypotenuse.
And because we need to find cosine u,
we will have to find x or the length of this adjacent side
which will be positive because we are in quadrant four.
So to find x, we use the Pythagorean Theorem
or the equation x squared plus y squared equals r squared,
which gives us x squared plus the square of negative 0.632
must equal one squared.
Solving for x squared, we have x squared equals
one squared is one
and then we subtract this quantity on both sides,
which gives us minus the square of negative 0.632.
And now we take the square root
of both sides of the equation.
We have a plus or minus here,
but because we are in quadrant four,
we know x will be positive.
Simplifying, we have x,
and now I'll go to the calculator
to get a decimal approximation
for this positive value of x.
We have the square root of one minus
the square of negative 0.632,
enter.
And x is approximately 0.774968.
Let's also label this on the reference triangle.
And now we have all the information we need
in order to find sine of u divided by two.
Applying the half-angle identity,
we have sine
of u divided by two is equal to
plus or minus the square root
of the quantity one minus cosine u
divided by two.
The next step is to determine whether we have
a positive or negative sine function value
for sine of u divided by two.
Because angle u is in quadrant four,
we know angle 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.
Because y is positive in quadrant two,
sine of u divided by two is positive.
So we have a positive square root
of the quantity one minus 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 0.774968 divided by one
or just 0.774968.
And this difference is divided by two
under the square root.
Now I go back to the calculator
to get a decimal approximation for this square root.
And we are told to round to four decimal places.
So we have the square root.
And we need this numerator in parentheses,
so we have open parenthesis one minus 0.774968,
closed parenthesis, divided by two,
still under the square root, and then enter.
Cosine of u divided by two is approximately 0.3354.
I hope you found this helpful.