Hi, I'm Dr Ian van Loosen from the Department of Mathematics and Statistics at Curtin University.
Let's consider the following situation.
Two boats leave the port at the same time, deviating away from each other at an angle
of 50 degrees. The first boat moves a total distance of 12 kilometre, the other boat moves
a total distance of 18 kilometres.
What we would like to know is how far those boats are away from each other. Situations
such as these can be solved by using the cosine rule.
We use the cosine rule when we are given either three sides of a triangle, and no angles.
Or two sides of the triangle and the angle enclosed between the two.
So for a general triangle - A, B, C - the cosine rules tells us that this side length
a squared must equal to side length b squared, plus c squared, take away twice their product,
multiplied by the cosine of the angle between the two.
Cosine rule also tells us that this side length b squared must equal to the sum of a squared,
c squared, take away twice their product, multiply it by the cosine of the angle between
those two sides.
And it also tells us that c squared must equal to a squared plus b squared, subtract away
twice their product, multiply by the cosine of the angle between those two sides.
So cosine rule - side of a triangle squared must equal to the sum of the two other sides
of the triangle squared, subtract away twice their product, multiply the cosine of the
angle between the two.
These three equations will allow us to solve for a triangle if I'm given two sides and
an enclosed angle. If I now go ahead and rearrange these equations for cosine of A, cosine of
B and cosine of C, I get the following three equations.
These three equations here will allow us to determine the angle in a triangle if I'm given
all the sides of the triangle and no angles.
So now lets go a head and look at the previous example.
So as we can see in this scenario, I've got a triangle, I've got two side lengths and
an angle which is enclosed between the two sides. So therefore I can apply
the cosine rule.
I'm trying to work out this side length here, the distance between the two boats. So using
the cosine rule. The cosine rule tells us, that this side squared must equal to the sum
of the other two sides squared. Subtract away twice their product. Multiply it by the cosine
of the angle between those two sides.
So from our calculator.
You work out the side length d squared. I'm after d, so simply take the square root of
this number.
Giving us the distance between the two boats of 13.80 kilometres.
So here is an example where I had a triangle, I had two side lengths, and an enclosed angle.
Lets look at one more situation.
In this situation I have the three airports, Adelaide, Brisbane and Canberra. I'm given
the distance between Adelaide airport and Brisbane airport of 1,600 kilometres. The
distance between Adelaide airport and Canberra as 1,000 kilometres and the distance between
Canberra and Brisbane is 900 kilometres. What we'd like to determine is the angle between
the two routes from Adelaide airport.
So in this case I want the angle between A and B and A and C.
Here I have a triangle, I have all sides in the triangle but I don't have any angles.
So once more I can use the cosine rule. The cosine rule tells us, to work out the cosine
of the angle at A. That is simply equal to the sum of b squared, c squared, take away
a squared, on twice the product of b and c.
So once more substituting the given information.
Evaluating this, tells me the cosine of the angle at A.
Now to evaluate for the angle at A, simply take the cosine inverse, of 0.8594, giving
us a value of 30.75 degrees.
So therefore the angle between the two routes from Adelaide airport is approximately
31 degrees.
So here was an example that had a triangle, three given sides, but no angles.
So in summary you need to use the cosine rule if you're given a triangle with all sides
and no angles or you're given a triangle with two sides and an enclosed angle.
Thanks for watching.