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

 

1. Law of Sines
We need the Law of Sines to help us find sides or angles that are not necessarily right triangles. Since the Pythagorean Theorem can only be used for right triangles, the Law of Sines helps us find the sides for triangles that are not right triangles.

 

http://etc.usf.edu/clipart/36700/36740/tri19_36740_lg.gif

To derive the Law of Sines, we first need to draw an imaginary line down angle B.




http://etc.usf.edu/clipart/36700/36738/tri17_36738_lg.gif



This imaginary line can be labeled h, as the picture above shows. This allows for two right triangles to be formed. We can now use SOH CAH TOA to help us figure out the rest (just SOH in this case)

Sin A=h/c

Sin C=h/a

Since both of these have h as a common variable, we can simplify to get c sin A=a sinC
Then, divide by ac and then you will get sinA/a=sin C=c.
(this applies to angle B too, if a perpendicular line was drawn from either angle A or C.)

You finally get:

 

 http://www.clarku.edu/~djoyce/trig/lawofsines.jpg



4. Area Formulas
The area of an oblique triangle is derived from the area formula which is A=1/2bh

http://www.compuhigh.com/demo/lesson07_files/oblique.gif

 

To find the height, a verical line is drawn and labeled h. The trig functions, which are sinA=h/c, sinB=h/b and sinC=h/a can be simplified by their denominators in order to make them equal to h. Then, this is plugged into the area of a triangle. H is then replaced with either asinC or csinA. Finally, you end up with the area of an oblique triangle, which is:

 

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZrTRmOJrogy3QUkgKj4h6OGuoWqoGscJ0J0Vx9qiKdr2IOoRiD67ngtI6YPhH06z0jxjEKdDKrUYCPBh8lVllMYlDLUvRHJ3G3KjA0Mm_OFJp-z-5Au4VdDeRht56fsINaXJ00Sijqu4/s400/hi.bmp

 

The area of an oblique triangle related to the area I am familiar with because it is what was used to derive the area of an oblique triangle. The area of a triangle is essential in finding the area of an oblique triangle because you need to plug the height into the familiar area equation in order to derive the area of an oblique triangle.