Welcome to a lesson on the law of cosines.
In this lesson we will derive the law of cosines and also
solve triangles using the law of cosines.
A triangle that is not a right triangle is called an oblique
triangle, until now we have only been able to solve
right triangles.
However, the law of cosines will allow us to solve
oblique triangles and solve new application problems.
However, in order to apply the law of cosines, we must
be given either a side angle side meaning two sides
and an included angle are known or side side side meaning
all three sides are known.
Oblique triangles with the following information can be
solved using the law of sines.
Side angle angle or angle side angle as well as side
side angle.
And now let's take a look at the law of cosines.
For any triangle with angles capital A, B, and C, and sides
lowercase a, b, and c, the law of cosines can be
expressed in any three of these ways.
And there are some patterns here in the equations
that should make them easier to remember.
Looking at the right side, whatever angle is used
in the cosine function, the opposite side is used
on the left side of the equation and now focusing on
the right side of the equation, notice that we have
b squared plus c squared we have b times c here.
If we have a squared plus c squared we have a times c here
and then if we have a squared plus b squared we have a
times b here.
Recognizing these patterns should make it easier
to remember the law of cosines and now let's derive
the law of cosines.
Beginning with an oblique triangle abc we construct
an altitude from let's say vertex c which forms a right
triangle on the left and on the right, where if we label
the leg of the right triangle on the right side x,
since the length of side ab is c, the leg in the other
right triangle the right triangle on the left would have
to be c minus x which is already labeled here.
And now to begin our proof, we focus on the right triangle
on the left and apply the Pythagorean Theorem
which gives us b squared equals y squared plus the quantity
c minus x squared, which is here in the proof.
The next step is to expand the square of c minus x
which gives us c squared minus 2cx plus x squared.
Now for the next step with apply the Pythagorean Theorem
to the right triangle on the right side, which gives us
a squared equals x squared plus y squared shown here
on the right and notice how our equation does have y squared
plus x squared so now we substitute a squared for x squared
plus y squared, which gives us a third equation
we now know b squared is equal to a squared plus c squared
minus 2cx.
Now using the triangle on the right again, notice the cosine
of angle b is equal to the ratio of the adjacent assigned
to the hypotenuse which gives us cosine b equals x
divided by a.
If we multiply both sides by a, we have x equals a cosine b
so now we perform a substitution for x we substitute a
cosine b for x, which gives us the fourth line and now
simplifying we have b squared equals a squared plus
c squared minus 2ac times cosine b which is one of the forms
of the law of cosines.
And using a similar process we could arrive at the other two
forms of the law of cosines.
Now that we have the law of cosines, let's look at some
examples.
We're asked to solve triangle abc if angle b is 73.1 degrees
side a is 24.2 feet and side c is 43.7 feet.
Let's label the information on the triangle.
Notice how I do have side angle side and therefore we can
use the law of cosines.
And because we have angle b we know we can find the length
of side b using the second form of the law of cosines.
This is why it's important to remember that whatever angle
we use on the right side, the opposite side is on the left
side of the equation, which means b squared is equal to a
squared plus c squared, which is 24.2 squared plus 43.7
squared minus two times a times c times cosine b
gives us minus two times 24.2 times 43.7 times cosine
of 73.1 degrees.
And now we need to go to the calculator.
The first thing to do is make sure we are in degree mode
by pressing the Mode key and notice on the third row
we are in the green mode.
So going back to the home screen, we now enter the right
side of the equation.
And press Enter.
This is a value of b squared, so this is a value
of b squared, let's round this to four decimal places,
but then we'll also store it in the calculator so we can
have the exact value.
So we have 1880.4716 to four decimal places.
Now to find b we take the square root of both sides
of the equation and go back to the calculator, but if we
take the square root of a rounded value we'll have
more of an error, so let's go ahead and store this
by pressing Store, let's store it in b by pressing Alpha x
Enter and now we'll take the square root of b to get a more
exact value.
So we'll press 2nd X squared Alpha Apps and Enter.
This will be more accurate than taking the square root
of 1880.4716 even though if we round the result
to one decimal place, in both cases we get b
as approximately 43.4 feet.
Let's go ahead and label this.
And now that we have an angle the length of the opposite
side we can use the law of sines to determine
the measure of angle a or the measure of angle c.
Let's use the law of sines and determine the measure
of angle a.
Using the law of sines we have the sine of angle a
divided by the length of the opposite side which is 24.2
must equal the sine of the known angle which is the sine
of 73.1 degrees divided by the length of the opposite side
which is approximately 43.4.
Now let's cross multiply and solve for sine a and then solve
for angle a.
Cross multiplying we have 43.4 times the sine of angle a
equals 24.2 times the sine of 73.1 degrees.
Now we divide both sides by 43.4.
Simplifying we have the sine of angle a is equal to
the quotient on the right.
Now to solve for a we take the inverse sine or [mumbles]
of both sides of the equation, which gives us angle a
is equal to the inverse sine of the quotient.
And now let's go back to the calculator.
And when we enter 43.4 we should use the more accurate
value rounded to four decimal places or the square root
of b.
So we'll press 2nd Sine for inverse sine and then we need
the numerator in parentheses so we have open parenthesis
24.2 times the sine of 73.1 degrees closed parenthesis
for the sine function, closed parenthesis for the numerator.
Then we need to divide this by either 43.3644
or the square root of b.
I'm going to use the square root of b so 2nd x squared
Alpha Apps right arrow, closed parenthesis and Enter.
So angle a is approximately 32.3 degrees.
Now from here we know the sum of the interior angles
of any triangle is equal to 180 degrees and if our angle c
is equal to 180 degrees minus the sum of 73.1 degrees
and 32.3 degrees, which is 74.6 degrees.
And now the triangle is solved.
Let's take a look at a second example.
In this example we're given the lengths of all three
sides, let's begin by labeling each side.
And now let's determine the measure of the largest angle,
which in our case would be angle c because angle c
is opposite the longest side.
Which means we'll be using this version of the law
of cosines where angle c is on the right side
of the equation.
C squared is 59.3 squared equals a squared plus b squared,
which is 25.4 squared plus 42.8 squared and then minus two
times a times b times cosine c which is two times 25.4
times 42.8 times cosine c.
And now I need to solve this equation for cosine c
so let's first subtract 25.4 squared and 42.8 squared.
This will give us 59.3 squared minus 25.4 squared
minus 42.8 squared equals negative two times 25.4
times 42.8 times cosine c.
And now to solve for cosine c we divide both sides
by this product.
Simplifying this quotient is equal to one giving us
cosine c is equal to the quotient on the left.
Now to solve for c we take the inverse cosine on both sides.
Simplifying on the right side we now have angle c.
Angle c is equal to the inverse cosine of this quotient.
Now we go to the calculator.
We enter 2nd Cosine for inverse cosine, and then
we need the numerator and denominator in parentheses.
So we have open parenthesis and the numerator.
Closed parenthesis and then divided by the denominator
in parentheses.
Open parenthesis, the denominator.
Closed parenthesis for the inverse cosine and Enter.
Rounding to one decimal place angle c is approximately
118.6 degrees.
But let's store this exact angle in the variable c.
We do this by pressing Store and then Alpha Program for c
and then Enter.
So again we have c is approximately 118.6 degrees.
Let's go ahead and label this on the triangle
and now because we have an angle and the length
of the opposite side, we can use the law of sines
to determine the measure of angle a or the measure
of angle b.
Let's use the law of sines to determine the measure
of angle a.
Using the law of sines we have the sine of angle a
divided by the length of side a which is 25.4
must equal the sine of the known angle which is
approximately 118.6 degrees.
We'll use the exact measure when we go to the calculator
and then divided by the length of the opposite side,
which is 59.3.
And now let's go ahead and cross multiply and solve
for sine a.
This gives us 59.3 times sine of angle a is equal to 25.4
times the sine of 118.6 degrees.
Now we divide both sides by 59.3.
Simplifying we have sine a is equal to the quotient
on the right.
And now we take the inverse sine of both sides
of the equation to solve for angle a.
Which gives us the measure of angle a is equal
to the inverse sine of this quotient.
And now going back to the calculator.
We enter 2nd Sine for inverse sine, and then we need
the numerator in parentheses, so we have open parenthesis
25.4 times the sine, now we don't want to use 118.6.
We could use this value here to four decimal places,
because we have the exact value in c we will press
Alpha Program for c, closed parenthesis for the sine
function, closed parenthesis for the numerator and then
this is divided by 59.3 closed parenthesis and Enter.
Angle a is approximately 22.1 degrees.
Now we can find the measure of angle b because we know
the sum of the interior angles must be 180 degrees
and therefore angle b must be equal to 180 degrees
minus the sum of 22.1 degrees and 118.6 degrees.
Actually this is an approximation because these are rounded
values, which gives us 39.3 degrees for the measure
of angle b and now the triangle is solved.
I hope you found this helpful.