Solve the triangle ABC
if angle A is 33.7 degrees, side b is 9.1 yards,
and side c is 13.1 yards.
Let's begin by labeling the angles
and the sides of the triangle.
We use capital letters for angles,
and lowercase letters for sides.
Because angle A is an acute angle of 33.7 degrees,
let's label this angle A, which means the opposite side,
this side here, must be side a, let's label this angle B,
and therefore this is side b, and let's label this angle C,
and therefore the opposite, this side here, must be side c.
And now let's label the known information.
We know the measure of angle A is 33.7 degrees.
Let's also record this in the table.
We know side b has a length of 9.1 yards,
and we know side c has a length of 13.1 yards.
We need to find the measure of angle B and angle C,
as well as the length of side a.
Because we don't have a known angle,
and a length of an opposite side,
we cannot use the law of sines yet,
but we do have a known angle,
and the length of the two sides that form that angle,
and therefore we can use the law of cosines.
We can use the law of cosines to find the length of the side
opposite the known angle, in this case, side a.
Because we will use the law of cosines,
to find the length of side a,
we'll begin by using the first equation,
a squared equals b squared plus c squared
minus two times b times c, times cosine of angle A.
Performing substitution, we have a squared
equals b squared, which is 9.1 squared,
plus c squared which is 13.1 squared,
minus two times b times c times cosine A,
is minus two times 9.1,
times 13.1,
times cosine of 33.7 degrees.
And now to solve for a, we take the square root
of both sides of the equation,
and because we know a is a length, and a is positive,
we only take the principle square root,
or positive square root, of both sides,
which gives us a is equal to the square root,
of 9.1 squared,
plus 13.1 squared,
minus two times 9.1 times 13.1
times cosine of 33.7 degrees.
And now we'll go to the calculator,
and get our decimal approximation, for the length of side a.
It is important to make sure the calculator
is in degree mode, notice how it says degree here.
So, now we enter secant x squared for the square root,
and then we have 9.1 squared, plus 13.1 squared,
minus two times 9.1,
times 13.1,
times cosine, 33.7 degrees,
close parenthesis, and enter.
Let's first round a to four decimal places.
Notice how there's a 8 in the fifth decimal place,
and therefore we round up, a is approximately 7.4877.
When giving the value of a, we will round
to one decimal place, just like b and c,
so we will say that a is approximately 7.5 yards.
But to make sure our future calculations are more accurate,
we will use a rendered to four decimal places,
and therefore let's record this value of a, on the triangle.
We'll say a is approximately 7.4877 yards.
Now that we have the measure of an angle,
and the length of the opposite side,
we can use the law of sines to find the measure of angle B,
or the measure of angle C.
But notice how angle C is opposite the longest side,
and therefore angle C might be an obtuse angle.
Remember, finding an obtuse angle, using the law of sines
is more challenging, so it's best to leave the obtuse angle
for last, let's find the measure of angle B,
using the law of sines.
Using the law of sines, we will use the sine
of angle B, divided by the length of side b,
is equal to the sine of angle A,
divided by the length of side a.
Notice how of these four values, only one is unknown,
which should be the measure of angle B.
So, we have the sine of angle B,
divided by 9.1 yards,
must equal the sine of A,
which is the sine of 33.7 degrees,
divided by the length of side a,
which we will approximate as 7.4877.
If we use 7.5 here, we will have more of an error,
when determining the measure of angle B.
So the next step, let's multiply both sides by 9.1.
Simplifying, 9.1 divided by 9.1 simplifies to one,
giving us sine B,
equals, on the right side we have 9.1 sine,
33.7 degrees,
divided by, 7.4877.
Remember our goal here, is to solve for angle B,
not sine B, to undo the sine function and solve for angle B,
we take the inverse sine of both sides of the equation.
Inverse sine of sine B, simplifies to angle B,
angle B is equal to the inverse sine of this quotient,
which we will approximate on the calculator.
We press secant sine for inverse sine,
and we need 9.1 times sine 33.7 degrees in the numerator,
so we need an open parenthesis for the numerator,
and then 9.1,
sine, 33.7,
close parenthesis for the sine function value,
close parenthesis for the numerator,
and then divided by 7.4877,
close parenthesis for the
inverse sine function, and then enter.
Let's first round angle B to four decimal places,
because we have a four in the fifth decimal place,
we round down, angle B is approximately
42.4011 degrees.
But because angle A is to one decimal place,
let's round angle B to one decimal place as well,
let's give angle B as 42.4 degrees.
And now, to find the measure of angle C,
we will use that the fact that the sum
of the interior angles of any triangle,
is equal to 180 degrees.
Measure of angle A, plus the measure of angle B,
plus the measure of angle C, must equal 180 degrees,
which means the measure of angle C,
which we are looking for, is equal to 180 degrees,
minus the measure of angle A, minus the measure of angle B,
which gives us the measure of angle C is equal to
180 degrees, minus 33.7 degrees,
minus 42.4 degrees,
and therefore the measure of angle C, is equal to,
or is approximately, 103.9 degrees.
Before we go, one quick check is to make sure
that the longest side is opposite the largest angle,
and the shortest side is opposite the smallest angle.
Notice how the longest side of 13.1 yards,
is opposite the largest angle of 103.9 degrees,
and the shortest side, of approximately 7.4877 yards,
is opposite the smallest angle of 33.7 degrees.
I hope you found this helpful.