We're asked to use the law of sines
to solve the triangle ABC if angle A equals 48 degrees,
side a has a length of four units,
and side b has a length of five units.
Notice here we are given side, side, angle.
When using the law of sines
and we're given side, side, angle,
there are three possibilities.
We can have no solution, one solution,
or two solutions.
We will discuss this
as we go through the solution process.
For the initial setup, let's use this triangle here
where angle A is 48 degrees
and the opposite side, side a, is four units.
Let's let this be angle B, and therefore the opposite side,
side b, is five units.
Let's let this be angle C, and this will be side c.
Let's begin by determining the measure of angle B.
Using the law of sines
we have the sine of 48 degrees divided by four
must equal the sine of B divided by five.
Now we could cross multiply here,
but let's go ahead and multiply both sides of the equation
by five to solve for sine B.
If we multiply both sides by five,
we get five times sine of 48 degrees divided by four
equals sine B.
Now we need to solve the equation for B
by taking the inverse sine of both sides of the equation.
This gives us inverse sine
of five sine 48 degrees divided by four equals B.
When we take the inverse sine of sine B,
it simplifies perfectly to B.
Now we need to go to the calculator
and get a decimal approximation for the measure of angle B.
It's important to make sure the calculator
is in degree mode.
And then we press second sine for inverse sine.
We need parenthesis around five sine 48 degrees
to make sure your product is in the numerator.
So we have open parenthesis five sine 48,
closed parenthesis for the sine function value,
closed parenthesis for the numerator
and then divided by four,
closed parenthesis for the inverse sine function, and enter.
At this point if we were to get an error,
we would know that there's no solution
or no possible triangle from the given information.
We would also round to the nearest tenth.
Because there's a six in the hundredths place value,
we round up.
B is approximately 68.3 degrees.
So because we found one value for B,
we know there is at least one solution.
Let's check to see if there are two solutions.
Remember the sine function value is positive
in the first quadrant and the second quadrant,
and therefore, there's an obtuse angle
in the second quadrant that also has
this same sine function value.
Let's find that angle and see if would work
for this triangle.
So again, the angle we found
on the calculator was 68.3 degrees.
Let's say this angle here.
To find the obtuse angle in the second quadrant,
we sketch a reference angle of 68.3 degrees
in the second quadrant, which would look something like this
and therefore the obtuse angle
that has the same sine function value
would be this angle here.
So another possibility for angle B
would be approximately 180 degrees, or half a rotation,
minus the reference angle of 68.3 degrees.
And therefore B could also be approximately 111.7 degrees.
To see if we have second solution, though,
we need to make sure that sum of angle B and angle A
is less than 180 degrees.
If it's greater than or equal to 180 degrees,
we don't have a second solution.
If it's less than 180 degrees, we do have a second solution.
Remember the sum of the interior angles of a triangle
is always equal to 180 degrees.
Well because 111.7 plus 48 is less than 180 degrees,
in this case we do have a second solution.
So when B is acute, or 68.3 degrees,
the triangle looks something like this,
where B is an acute angle,
but if we swing side a to the left,
notice how B is now an obtuse angle,
which would measure 111.7 degrees.
So we do have two solutions here.
We need to solve both of these triangles.
So the next step, let's find the measure of angle C
in both triangles.
So for the first triangle where all the angles are acute,
angle C is going to be approximately,
because this angle is rounded,
180 degrees minus the two interior angles
that we already know,
so minus 48 degrees minus 68.3 degrees
and therefore C is approximately 63.7 degrees.
And now for the second triangle where angle B is obtuse,
we have C is approximately 180 degrees
minus the two known interior angles,
which is minus 48 degrees minus 111.7 degrees.
So for the second triangle C is approximately 20.3 degrees.
Now, now for both triangles
we need to find the length of side c.
Going back to the first triangle,
let's be sure to use the angle of 48 degrees
since this angle is exact.
Let's begin with the sine of 48 degrees divided by four
must equal the sine of 63.7 degrees divided by C.
And now we could cross multiply,
but let's go ahead and multiply both sides of the equation
by the least common denominator, which is 4C.
Simplifying before multiplying, four divided by four
simplifies to one, giving us C sine 48 degrees equals,
on the right side C divided by C simplifies to one,
giving us four sine 63.7 degrees.
To solve for C, we divide both sides by sine 48 degrees.
Simplifying sine 48 degrees divided by itself
simplifies to one.
And now we go to the calculator
and get our decimal approximation for C
in the first triangle.
Again, we need the numerator in parenthesis.
We have open parenthesis four sine 63.7
divided by sine 48 degrees and enter.
To the tenths place value, C is approximately 4.8.
And now we need to find C in the second triangle,
where angle B is obtuse.
We'll begin with sine 48 degrees divided by four
equals in this triangle we have
sine 20.3 degrees divided by C.
Again we multiply both sides by 4C.
Simplifying before multiplying.
On the left we have C sine 48 degrees equals on the right
we have four sine 20.3 degrees.
To solve for C, we divide both sides by sine 48 degrees.
Simplifying, sine 48 degrees divided by itself
simplifies to one.
So now we know C is equal to this quotient,
which we'll go back to the calculator
and get a decimal approximation.
The only difference is, we need to change 63.7 to 20.3.
So if we press second enter, it brings up the previous entry
and we can now edit the entry by using the arrows
and overwriting any value we want.
So let's replace 63.7 with 20.3 and press enter.
And to the tenths place value
we have C is approximately 1.9.
So in the second triangle, C is approximately 1.9.
Let's go ahead and summarize our results.
We know we have two solutions.
So both triangles we know angle A is equal to 48 degrees.
In the first triangle, angle B is approximately 68.3 degrees
and angle C is approximately is 63.7 degrees.
In the second solution,
angle B is approximately 111.7 degrees.
Angle C is approximately 20.3 degrees.
And then for both triangles we know a is equal to four units
and b is equal to five units.
But for the first solution, our first triangle,
c is approximately 4.8
and for the second solution, c is approximately 1.9.
I hope you found this helpful.