BQ#1: Unit P Concepts 1 and 4: Law of Sines, Area of an Oblique Triangle, and Derivations

Law of Sines:

What's the formula?

 

Why do we need such  a thing?
- We need the Law of Sines to help us solve for triangles that are not right triangle (that do not have a 90 degree angle). For a normal right triangle it would be simple to solve for them since we could just Pythagorean Theorem( a^2+b^2=c^2) or we could use other simple formulas to find there angles but, with a non-special right triangle we could use the Law of Sines formula presented above.

 How do we know it is already derived from things we know?

http://library.thinkquest.org/C0121962/sincos1.gif

- If we were to take a scalene triangle, and just draw a line from the top to the base of the triangle we would get two right triangles, shown as following. Once having drawn that, the picture would look familiar because thanks to Sin, Cos, Tan, and their inverses we would be able to solve them. A perfect example would be : sinA=h/c and sinC=h/a.

http://dradqrf3hyj25.cloudfront.net/thumbnails/Alg2_13_02_0004-diagram_thumb.png

If you were to set both example ratios equal to each other you will realize that both ratios have "h" in common and if you cancel that you would get a whole different ratio, as show below.

If you were to take a look at the triangle again, you will see the m<B could form another equation.  Another example: sinB=h/c sinC= h/b. If you do the same thing as the previous ratio you notice that they both have h in common as well, so preform the same thing and you will conclude that  sinB/b= SinC/c.

Area of an Oblique Triangle:

How is the “area of an oblique” triangle derived? 

If you look at the picture shown on the right, you will see an oblique triangle and the height being drawn to it. Then knowing what we saw above, that sinC=h/a, sinA=h/c, and sinB=h/a(c) we can then multiply them so they can equal "h" and once that is done we can use the are of triangle formula know as A=1/2bh. With that being said, we can then substitute the bh for things we know such as sinB, sinC, sinA. 

How does it relate to the area formula that you are familiar with? 

Well we were familiar that the area of a triangle is A= 1/2 bh, and knowing that we can actually replace the "h" with values such as sinB, sinC, sinA.