BQ #1: Unit P Concept 1 & 4 - Law of Sines and Area Formula

1. Law of Sines - Why do we need it?  How is it derived from what we already know?

The law of sines is needed to figure out the missing parts (angles and sides) of a non-right triangle. Non-right triangles are more complex to solve for since they are not like special right triangles and have a special side relationships (ex: 45-45-90 triangle has a relationship of (x)-(x)-(x)radical 2).
The deviation to this law is shown through the process and pictures below:
First, there is the non-right triangle ABC. An imaginary perpendicular line is drawn to cut angle B. The imaginary line is labeled as h.

The perpendicular line created (2) right triangles. From there, the basic trig functions of sin (opposite/hypotenuse) can be used for angle A and C. The trig functions are then rearranged to equal to h.  

Since both functions are now equal to h, they can be equal to one another. Both are then over the corresponding sides used (sides a and c). Simplification can be done and the law is finished. (*Note: this implies the same for B if the perpendicular line was cut either from angle A or C)

4. Area formulas - How is the “area of an oblique” triangle derived?  How does it relate to the area formula that you are familiar with?

The area of an oblique triangle is derived from the triangle area equation and a trig function. The area of any triangle is (1a) A=1/2bh, where b is the base and h is the height. However, in an oblique triangle, the height is not given and the sin trig function must be used.
With the oblique triangle provided as an example, triangle ABC is cut perpendicularly in half from angle B to create 2 right triangles with the sharing side of h. Side h must be used with sin, in this example sin C is used with h/a (sin A could have been used as well). (1b)The sin trig function is rearranged to equal h.

Since the sin trig function now equals to h, it can be substituted into the (1a) area of triangles. (*Note: the perpendicular line from the beginning can cut any angle and create in total 3 ways to find the area with the same concept).

     It relates to the formula we are familiar with since it is mostly substitution, the new equation has parts from the old one. The triangle is still multiplied by 1/2 and consists of 2 sides. However in the new one, it also consists of an angle which is in between the two sides used. In the original one, meanwhile, uses the base and the perpendicular height.