BQ #1: Unit P Concepts 1 & 5: Law of Sines and Area of Oblique Triangles

1. Law of Sines

Why do we need it? 

We use the law of sines because it helps us evaluate the side lengths or angles for triangles, in this unit mainly oblique triangles. It is a simple process of evaluating these missing sides by making to proportions based on what we know and what we need. The proportions you will need will be based on the law of sines formula given to the right.





How is derived from what we already know? 

First, we know that triangle ABC is oblique making it have different different side and angle lengths. Since it oblique, we are able to cut the triangle in a section where it makes two right triangles. (as shown on the top picture) This procedure will help us evaluate the height of the triangle as well.

Since we know that Sine is opposite/hypotenuse, we can use both angles A and C to make two trig functions. The two trig functions that came from this triangle are shown on the second picture. (The very top one)

The next step we must take is to cross multiply which makes h remain alone while we have cSinA and aSinC as the two trig functions. We want to have the Angle value the same as its corresponding side so we must divide both functions by ac. This will make the first equation get rid of the "c's" while the other will get rid of the "a's".

Finally, we get what we were looking for which is SinA/a = SinC/c









5. Area Formulas

Do all three area formulas work? 

As we can see on the pictures below, we notice that all three equations are basically the same the same answers. They all have a 12 unit area which proves that all three equations work to find the area of oblique triangles.