Why do we need it?
The law of sines is needed when solving for triangles. To find angles and lengths in right triangles,we use the Pythagorean Theorem and normal trig functions. However, when we are faced with a triangles that are not RIGHT triangles, we result to using the law of sines, which we can find using ASA or AAS.
How is it derived from what we already know?
We start off with a triangle that is not a right triangle.
We draw a perpendicular line down from angle A, forming two right triangles. Label the line h (for height).
Since we have two right triangles, we can use trig functions. We have to find h, and we know the equations are sinA=h/c and sinC=h/a in right triangles. To get h by itself, we multiply the denominators by itself for both equations. So the result is csinA=h and asinC=h. Since they are both equation to h, you can make them equal to each other. So it will become csinA=asinC. Next, we divide the equation by the coefficients together, ac. The result is sinA/a=sinC/c.
Using the same concept, we can now find angle B. Let's draw a perpendicular line through C.
With the two new right triangles, the trig functions are asinB=h and bsinA=h. Set them next to each other by the transitive property. Next, we again divide the denominators multiplied together, ab, and we get sinB/b=sinA=a.
The result: Law of Sines
Area of an Oblique Triangle:
How is the "area of an oblique triangle" derived?
Our final formula for area of an oblique triangle
How does it relate to the area formula that you are familiar with?
We used the original area formula a=1/2bh to find the area of the oblique triangle. The only difference is that we had to find the height using trig function sine, instead of normally being given it. The rest of the area formula is the same.
(and beautiful triangles drawn by me!!! :D)