Question 392634
The Law of Cosines has four variables: the lengths of the 3 sides of a triangle and the measure of one angle.<br>
Your problem gives you 2 sides and the measure of one angle. So you know the values 3 of the 4 variables in the Law of Cosines. Anytime you know all but 1 variable in an equation you should be able to solve for the value of that last unknown. So we will use the Law of Cosines to solve this problem:
{{{(AC)^2 = (AB)^2 + (BC)^2 - 2(AB)(BC)cos(B)}}}
To convert BC into meters, we divide by 39.37:
BC = 423 inches = 432/39.37 meters = 10.7442214884429769 meters.
Also 132 degrees and 40 minutes = 132.67 degrees since 40 minutes is 2/3 of a degree and 2/3 as a decimal is approximately 0.67.
Inserting these values into our Law of Cosines equation we get:
{{{(259)^2 = (AB)^2 + (10.7442214884429769)^2 - 2(AB)(10.7442214884429769)cos(132.67)}}}
To avoid confusion I am going to replace (AB) with an "x":
{{{(259)^2 = (x)^2 + (10.7442214884429769)^2 - 2(x)(10.7442214884429769)cos(132.67)}}}
Simplifying we get:
{{{(259)^2 = (x)^2 + (10.7442214884429769)^2 - 2(x)(10.7442214884429769)(-0.6777747765428034)}}}
{{{67081 = x^2 + 115.4382953927198178 - (-14.5643246369116504x)}}}
{{{67081 = x^2 + 115.4382953927198178 + 14.5643246369116504x}}}
This is a quadratic equation because of the {{{x^2}}}. SO we want one side to be zero. Subtracting 67081 from each side we get:
{{{0 = x^2 + 14.8742038845055153x -66975.5617046072801800}}}
Now we can use the Quadratic Formula:
{{{x = (-(14.8742038845055153) +- sqrt((14.8742038845055153)^2 - 4(1)(-66975.5617046072801800)))/2(1)}}}
which simplifies as follows:
{{{x = (-(14.8742038845055153) +- sqrt(221.2419411978389608 - 4(1)(-66975.5617046072801800)))/2(1)}}}
{{{x = (-(14.8742038845055153) +- sqrt(221.2419411978389608 + 267902.2468184291207000))/2(1)}}}
{{{x = (-(14.8742038845055153) +- sqrt(268123.4887596269597000))/2(1)}}}
{{{x = (-14.8742038845055153 +- sqrt(268123.4887596269597000))/2}}}
In long form this is:
{{{x = (-14.8742038845055153 + sqrt(268123.4887596269597000))/2}}} or {{{x = (-14.8742038845055153 - sqrt(268123.4887596269597000))/2}}}
We can see that the second equation will give us a negative value. But the side of a triangle cannot be negative. So we will reject the second solution. The only solution we can use is:
{{{x = (-14.8742038845055153 + sqrt(268123.4887596269597000))/2}}}
Using a calculator to find the square root we get:
{{{x = (-14.8742038845055153 + 517.8064201606880036)/2}}}
{{{x = 502.9322162761824883/2}}}
x = 251.4661081380912442
So AB is approximately 251.5 meters long.