Given the triangle,
find the length of angle A using the Law of Cosines.
Let's begin by labeling the angles
and the sides of the triangle.
We normally use capital A, B, and C for the angles,
and lowercase a, b, and c for the sides.
Notice angle A is already labeled here.
Let's also place a capital A outside the triangle.
Because this is angle A, the opposite side,
this side here must be side a that has a length of 17 units.
It does not matter which angle we label angle B or angle C.
Let's label this angle B,
and therefore the opposite side must be side b.
Side b has the length of 12 units
which means this is angle C,
and therefore the opposite side is side c
with a length of 23 units.
Because our goal here is to find the measure of angle A,
we will use the first equation for the Law of Cosines
where the right side of the equation has cosine of angle A.
So again, we will be using the equation
a-squared equals b-squared plus c-squared
minus two times b times c times cosine of angle A.
And now, we substitute 17 for a, 12 for b, and 23 for c.
Performing this substitution gives us
17 squared equals 12 squared plus 23 squared
minus two times 12 times 23 times cosine of angle A.
And now let's begin simplifying the equation.
On the left, 17 squared equals 289,
equals, 12 squared is 144 plus 23 squared is 529.
And then we have minus two times 12 times 23 times cosine A.
Since two times 12 times 23 equals 552,
we have minus 552 cosine angle A.
So simplifying, 144 plus 529 equals 673,
giving us 289 equals 673 minus 552 cosine of angle A.
We will now solve the equation for cosine A,
and then solve the equation for angle A.
To solve for cosine A, the first step
is to subtract 673 on both sides of the equation.
Simplifying, this difference is zero.
On the left, 289 minus 673 is equal to negative 384.
We have negative 384 is equal to
negative 552 times cosine angle A.
Next step is you divide both sides by negative 552.
Notice this quotient is one.
On the right side of the equation,
we now just have cosine of angle A.
On the left side,
a negative divided by a negative is positive.
Let's leave it as a fraction as positive 384
divided by positive 552.
This does simplify, but we'll go ahead
and leave it in this form.
Now looking at the right side, we have cosine A,
but our goal is to find the measure of angle A.
To undo the cosine function, we now take the
inverse cosine of both sides of the equation.
So we take the inverse cosine of the right side,
and the inverse cosine of the left side.
On the left side of the equation,
inverse cosine of cosine A simplifies to angle A,
so we have angle A is equal to
inverse cosine of this quotient which we will now use
the calculator to get a decimal approximation.
Going to the calculator, the first step is
to make sure the calculator is in degree mode
because the question does ask for degrees,
and notice how we are in degree mode here,
but to check, we press the Mode key,
and notice how degree is highlighted.
If degree was not highlighted,
we would arrow down to degree,
and then press Enter.
To go back to the Home screen,
press second Mode for quit.
To enter inverse cosine or arc cosine,
we press second cosine,
and then we have 384 divided 552,
close parenthesis and then Enter.
And again, if we round to the
hundredth place via two decimal places,
angle A is approximately 45.92 degrees.
I hope you found this helpful.