Given the triangle, find the length
of side x using the Law of Cosines.
Let's begin by labeling the angles and
sides of the triangle.
We typically use capital A, B and C for the angles
and lowercase a, b and c for the sides.
Let's label the known angle of 40 degrees angle A.
And therefore the opposite side,
it's a length of x units, would be side a.
We call this angle B, the opposite side is side b
with a length of 22 units
and if we call this angle C, then side c
has a length of 29 units.
Because we are looking for the length of side a,
we will use the first equation for the Law of Cosines
where we have a squared on the left side of the equation
and the known angle, angle A on the right side.
Notice for all the equations,
whatever side we have on the left side of the equation,
we have the opposite angle on the right.
So for our setup, we will use the equation
a squared equals b squared plus c squared
minus two times b times c times cosine A.
And now we substitute 22 for b,
29 for c and 40 degrees for angle A.
This gives us a squared equals 22 squared
plus 29 squared minus two times 22 times 29
times cosine 40 degrees.
And now let's begin simplifying
the right side of the equation.
22 squared is equal to 484
plus 29 squared is equal 841
minus two times 22 times 29 times cosine 40 degrees
equals 1,276 cosine 40 degrees.
And then 484 plus 841 equals 1,325,
giving us a squared equals 1,325
minues 1,276 cosine 40 degrees.
Notice to solve for a, we need to take the square root
of both sides of the equation
so we do not want to get a decimal approximation
for the right side, take the square root
and then round again because that will give us
more of an error.
So leave the right side of the equation in this form
and to solve for a, we will take the
square root of both sides.
Because a is a length, we only take the
principle square root, or positive square root
which gives us a is equal to this square root
which we will use the calculator
to get a decimal approximation.
And I believe our homework asks to round
to four decimal places.
So going to the calculator,
the first step is to make sure the calculator
is in the green mode.
Notice how right here it does say degree
but just to check, we can press Mode
and notice in the fourth row, degree is highlighted.
If degree was not highlighted,
we would arrow down to degree and press Enter.
To go back to the Home screen, we press second Mode
for Quit and then we enter the right side of the equation.
So we enter second x squared for the square root
and then we have 1,325
minus 1,276 cosine 40 degrees,
close parenthesis and Enter.
Notice in the fifth decimal place there's an eight,
which means you round up.
a is approximately 18.6421 units.
But notice originally, this side was labeled
with a length of x units
so now we know x is approximately 18.6421 units.
I hope you found this helpful.