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

1. Law of Sines 

Why do we need it?
We need the Law of Sines in order to solve for a triangle that are not right triangle. While for a right triangle we can just use the Pythagorean Theorem to find missing angles and sides. 

How is it derived and what we already know?

First lets look at a triangle, we know that it is not a right triangle because it does have a 90 degree angle label.













http://math.ucsd.edu/~wgarner/math4c/derivations/trigidentities/lawofsines.htm


To make a triangle a right triangle, we must draw a line from the angle C to line AB. The red line we call it perpendicular line and label it h, now we have 2 right triangles. We know that it is a right triangle because it has a square box, which means it is 90 degree angle.

Lets recall the area of the right triangle to be Area = 1/2 base times height. To find the area of the triangle, we know that sin A = h/c and that sin c = h/a. Then we divide them the first one by c and the second by a. To get csinA = h and asinC = h.





http://math.ucsd.edu/~wgarner/math4c/derivations/trigidentities/lawofsines.htm


To find angle B, we must make a perpendicular line from angle A to BC. Then use sinB = h/c and angle C is sinC = h/b. 

That will give us the formula for Law for Sines, which is: 

http://www.mathsisfun.com/algebra/trig-sine-law.ht

http://math.ucsd.edu/~wgarner/math4c/derivations/trigidentities/lawofsines.htm



4. Area of an Oblique Triangle

An oblique triangle is a triangle with all sides with different length. 

How is the area of an oblique triangle derived?
Since we know that sinC = h/a, sinB = h/a, and sinA = h/c. We multiply them by their denominator and you will get:



http://wps.prenhall.com/wps/media/objects/551/564474/StudyGuide/7c1h7_1.gif

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

How does it relate to the area formula that you are familiar with?
It basically uses the same formula, you just plug into the equation and find the area. We just need to use sine in order to find the height.


References:
http://www.mathsisfun.com/algebra/trig-sine-law.html
http://math.ucsd.edu/~wgarner/math4c/derivations/trigidentities/lawofsines.htm
http://wps.prenhall.com/wps/media/objects/551/564474/StudyGuide/7c1h7_1.gif
http://www.compuhigh.com/demo/lesson07_files/oblique.gif