It’s Professor Dave, let’s learn the law of sines.
We have learned a lot about trigonometric functions as they pertain to right triangles.
But what about other types of triangles?
Surely we can do something with these as well.
As it happens, there is a law that will help us find the sides and angles of oblique triangles,
as well.
Oblique triangles are ones that don’t have a right angle.
For these, all the SOHCAHTOA stuff we learned won’t work anymore.
But there are other relationships we can use, and one of them is called the law of sines.
This law says that for an oblique triangle with angles A, B, and C, all capitalized,
as well as sides A, B, and C, all lower case, where the sides are precisely opposite their
corresponding angles, the relationship between all of these will be A over sine A equals
B over sine B equals C over sine C.
This is pretty easy to remember, and not too hard to apply.
We just have to know a few of these values, and we will be able to solve for the rest.
Let’s look at this side angle angle triangle.
That means we know one side, and two of the angles.
We can get the rest of the sides and angles using the law of sines.
First, given two angles, we can always get the third, we just subtract these two angles
from one eighty.
Now, we have one angle-side pair, so we can use its ratio to get the other two sides.
Here’s one relationship, we just bring this angle up over here, and then evaluate.
To get the third side, here’s the other relationship, and we just do the same thing.
When we use this technique, we will usually measure angles in degrees, and just plug things
into a calculator to get approximate side lengths.
Now that we’ve seen a side angle angle example, let’s try angle side angle.
This is where the side we know is specifically the one in between the two known angles.
In such a case, we will again subtract the known angles from one eighty to get the third,
and once we have that, we are ready to plug values into the law of sines.
Let’s look at a slightly different scenario.
Let’s say we know two side lengths and one angle, which means side side angle.
Here, we have to first find the angle opposite this side length.
Once we have that, we can subtract these two angles from one eighty to get the third, and
find the third side length if we need to.
The interesting thing about this case, is that sometimes, given two sides and an angle,
the law of sines will prove that the third side can’t possibly connect the other two,
and the triangle can’t exist.
This would occur if we solve for the sine of another angle and this value exceeds one,
which is impossible for the sine function.
It is also possible that SSA triangles result in two solutions, if we get a value for sine
that could correspond to two different angles.
The last thing we want to look at is the formula for calculating the area of an oblique triangle.
We know that the area of any triangle is one half base times height, but if we don’t
know the height, we can also use this formula, where the area is equal to the product of
the lengths of two sides and the sine of the angle between them.
Since there are three angles, there are three ways to get this per triangle.
That’s pretty much all we need to know regarding the law of sines, so let’s check comprehension.