Given tangent u equals negative eight 7ths
in quadrant four, find the exact value of the following.
We'll determine the trig function values
using the half-angle identities shown here below,
before we do this though, let's show the reference angle
and reference triangle for angle u in quadrant four,
let's assume this is the terminal side
of angle u in quadrant four,
the reference angle is this acute angle here,
we'll call it u sub r,
the reference triangle is this triangle here,
from here we know that tangent u
is equal to the ratio of the opposite side
to the adjacent side or in the coordinate plane,
y divided by x, we are in the 4th quadrant,
where x is positive and y is negative
and because we know the tangent function value
is negative eight 7ths, let's write the tangent function
value as negative eight over positive seven,
which means we can label y
or the opposite side negative eight
and x on the adjacent side positive seven
and because we have to find the cosine of angle u,
we will now find the hypotenuse or r
using the Pythagorean theorem, which gives us r squared
is equal to seven squared plus the square of negative eight,
r squared is equal to 49
plus 64,
r squared is equal to 113
and then we take the square root of both sides,
r is always positive and therefore we only take
the principle square root or the positive square root,
r is equal to the square root of 113.
We are going to need the cosine of angle u,
so let's go ahead and find that now,
cosine u is equal to x divided by r
or the ratio of the adjacent side to the hypotenuse,
which is positive seven divided by the square root of 113.
And now let's find the sine of u divided by two,
remember the angle u terminates in quadrant four,
so let's assume that angle u is between 270 and 360 degrees,
notice how we divide everything by two,
we have the angle u divided by two
is between 135 and 180 degrees, which is in quadrant two,
this is important because in quadrant two,
x is negative and y is positive,
which means the cosine of u divided by two
is going to be negative, because x is negative
and because y is positive,
the sine of u divided by two is going to be positive,
so the sine of u divided by two
is equal the positive square root
of one minus cosine u, which is one minus
seven over the square root of 113 all divided by two.
To simplify this, instead of dividing by two,
we will multiply by the reciprocal, which is one half,
this is equal to the square root
of the numerator of one minus seven
divided by square root 113
and then instead of dividing by two,
we will multiply by the reciprocal of one half.
Distributing the one half, we have the square root
of one half times one is one half
minus seven over the square root of 113 times one half
is seven over two square root 113
and we'll go ahead and leave it in this form here,
this is the exact value of sine of u divided by two.
And now let's find the cosine of u divided by two,
which again, because the angle terminates in quadrant two,
the cosine function value is negative,
so we have negative square root of one
plus cosine u divided by two,
which is the quantity one plus seven
over the square root of 113 divided by two,
again, let's multiply the sum in the numerator
by the reciprocal of the denominator,
this is equivalent to negative square root
of the numerator of one plus seven over square root 113
times the reciprocal of two or times one half.
Distributing the one half, we have negative
and then the square root of one half times one is one half
and then we just have plus seven over two,
the square root 113,
this is the exact value of cosine of u divided by two.
And now let's find the tangent of u divided by two,
to do this, we can use any of these three identities here,
let's use this identity here,
where the tangent of A divided by two
is equal to the sine of A
divided by the quantity one plus cosine A.
So for us, the tangent of u divided by two
is equal to the sine of A, sine is equal to the ratio
of the opposite side to the hypotenuse,
which is negative eight over the square root of 113,
this is divided by one plus cosine A,
which is one plus seven
divided by the square root of 113.
To simplify this complex fraction,
let's multiply the numerator and denominator
of the complex fraction by the square root of 113.
So we will multiply this by the square root of 113
as well as the denominator,
which should clean this up a little bit.
This is equal to, in the numerator,
notice how the square root of 113
simplifies to one here and here,
leaving us with just negative eight,
we have negative eight over,
in the denominator, the square root of 113
times one is the square root of 113
and then for the second product,
the square root simplifies to one,
leaving us with just plus seven,
this is the exact value of tangent of u divided by two.
I hope you found this helpful.