8.1 The Law of Sines

Hi! So this section is about the Law of Sines. Its good if you have a triangle thats oblique, or a triangle without a right angle, which is most triangles in the world.



So here we have our basic arbitrary triangle with sides a, b, and c and angles α, β, and γ. We have also drawn an altitude, h. From here, we can derive the Law of Sines.


sin α= h/b, so h= b sin α


AND


sin β= h/a, so h= a sin β


If we set the two equations equal to eachother, we get
b sin α= a sin β, or (sin α)/a= (sin β)/b


If we draw another altitude, we could also prove that (sin γ)/c is equal to these two.




So... in the end this is what we get:

If you want to use this formula, you need to know a minimum of three parts of the triangle. You can use it when you know these parts:

1. SSA - two sides and the angle opposite one of them

2. AAS/ASA - two angles and any side

To find SAS or SSS, we will have to use the Law of Cosines instead.

Theres also something important you have to keep in mind when solving with SSA. Sometimes, you solve for the sine of an angle, and you get some number between 0 and 1. There are 2 angles between 0 and 180 that have that sine, because sine functions are positive in the 1st and 2nd quadrants. This is called the ambiguous case. 

So how do you decide which angle is right? The important thing to remember is that if the measure of angle α (or β or γ) is greater than 90 degrees, than side a>b (or b>c or c>a.) If this is not the case, than angle α is acute.

That's pretty much all we covered in this chapter, so sorry the blog is so short! Bye!