Hi, I'm Dr Ian van Loosen from the Department of Mathematics and Statistics at Curtin University.
Let's say I want to solve a triangle which is a non right angled triangle. Just as we
have here.
To solve a triangle such as this, we need to use either the sine rule or the cosine
rule, depending on what information is given to us.
Let's start by looking at the sine rule.
We use the sine rule whenever we are given a triangle with two angles and any given side
or two sides and any angle which is not enclosed by them.
So, for a general triangle - A, B, C - the Law of Sines tell us that a side length is
proportional to the sine of its opposite angle.
So, stated mathematically, we have: a, divided by the sine of its opposite angle A, must
equal to side length b, divided by the sine of its opposite angle B, equals the side length
c, divided by the sine of its opposite angle C.
The sine rule, as we can see, is made up of three equations. To solve for a triangle we
don't need to use all three equations.
So let's go back to our earlier example.
So here we can see in this case we have a triangle, we're given two side lengths and
an angle which is not enclosed by these two side lengths. So we can solve this by using
the sine rule.
So, we're given side length a, and its opposite angle capital A, and we're given side length
little b. So the first thing we can calculate is the angle at B.
So, from the sine rule. Sine rule says: a, divided by the sine of A equals, b divided
by the sine of B.
Putting in our given information: side length a, divided by the sine of its opposite angle
which is 21 degrees, equals to the side length b, divided by the sine of its opposite angle
capital B.
So making sine B the subject of our equation, what we'll do is multiple both sides by sine
of B, multiply both sides by the sine of 21 degrees but we'll also divide through by 14.
So we get: sine of B is equal to 17 times by the sine of 21 degrees divided by 14.
Evaluating this on our calculator.
We get the following value.
Now solving for this angle B, from our calculator, we simply take the sine inverse of our value,
0.43.
Our calculator will give us the angle of 25.8 degrees.
In this case we need to be a little bit careful. Cause there is a possible, a second possible
solution at this angle B.
That is, to get our second solution - from 180 degrees we subtract away that previous
angle.
And that will give us 154.20 degrees.
This is a second possible angle that we can have in this triangle.
The reason being, if we look at the graph of the sine function between 0 and 180 degrees,
there are always going to be two angles which are going to give us the same constant value.
For here, if I take the sine of 25.8 degrees and the sine of 154.2 degrees, they're both
going to give me the same constant value of 0.43.
What does this second angle tell us? Well simply due to limited information that was
given to us, there's going to be two distinct triangles which fit the given description.
In a practical problem you might actually be able to discount one of these angles. Also,
this is only a second possible solution, provided when we add it to the angles that we already
have in our triangle that they add up to less than 180 degrees.
Here if I add 154 degrees to 21 degrees, I get 175 degrees which is less than 180 degrees.
So this is a second possible solution.
So therefore in this example, here we will pursue both of the cases.
So, starting with our first case.
With the angle at B is equal to 25.8 degrees. So now we have two angles in our triangle.
To determine the last angle in our triangle, at C, all the angles in a triangle add up
to 180 degrees.
So 180 degrees, subtract away those two angles that I already have. Gives me my remaining
angle at C.
So now I have all the angles in the triangle. One bit of information I still haven't yet
found is the side length C.
To work out the side length C, again I can use the sine rule.
So the sine rule tells us that a, divided by the sine of A, must equal to side length
c, divided by the sine of its opposite angle.
So again, we put in the information that we know.
a, 14, divided by the sine of its opposite angle. Must equal to this side length c, divided
by the sine of its opposite angle. Which we just calculated to be 133.2 degrees.
Now, making this c the subject of the equation. I'll just multiply both sides by that sine
of 133 degrees.
So we're evaluating this, gives us our side length at little c.
So here we've solved our triangle, using the case when the angle at B is equal to 25.8
degrees.
Now lets look at the other case.
So, in our case two, is where our angle at B
was equal to 154.2 degrees.
So again, like last time, we have two angles in our triangle. I've got one remaining angle
to work out at C.
So subtracting these two angles, from 180 degrees.
Will tell me that the angle at C is equal to 4.8.
So once again the last bit of information we have to determine is that side length,
little c.
So we'll apply sine rule once more.
So the sine rule tells us: a, divided by sine of A, must equal to c, divided by the sine
of c.
Again, putting in our given information: side length a, 14, divided by the sine of its opposite
angle. Must equal to side length c, divided by the sine of its opposite angle, which we
just worked out.
Making c the subject of the equation. Multiply both sides by the sine of 4.8 degrees.
Evaluating this on our calculator. Gives us our side length at little c.
So here we've solved the triangle, in the second case where the angle at B was equal
to 154.2 degrees.
So in summary we use the law of sines if we're given a triangle where we have two side lengths
and an angle which is not enclosed between the two sides or we're given at least two
angles and one side.
However we can't use the sine rule if we were given a triangle with all side lengths but
no angles or we're given a triangle with two side lengths and the angle enclosed between
the side lengths. In situations such as these we need to use the cosine rule.
Thanks for watching.