BQ #1: Unit P Concept 3 and 4: Law of Sines and Area of an Oblique Triangle

Law of Sines:

1. Why do we need it? How is it derived from what we already know?

    The Law of Sines is used for non-right triangles. We cannot use the Pythagorean Theorem for non-right triangles. The Law of Sines can be used when it comes to non-right triangles with AAS and ASA. Remember, just like sine on the Unit Circle, the Law of Sines cannot be greater than 1 or less than 1. To make sure where the Law of Sines is derived from, draw a non right triangle.

   To make a right triangle, draw a perpendicular line from the top of the triangle to the bottom and name that "h." Now that you have two triangles, you can use SOH CAH TOA. When doing using the soh cah toa, you found that sinA= h/c and sinC= h/a. To get rid of "h", you multiply both sides by its denominator. The final answer is cSinA= aSinC. To get the Sine angle alone, divide both sides by "a" and "c".

Area of an Oblique Triangle:

1. How is the "area of an oblique" triangle derived?

   The area of an oblique triangle is derived when you substitute sinC=h/a, which is h=asinC, into the regular area equation of a triangle, A=(1/2)BH. By doing that, you will get 1/2b(asinC). Remember, we are still using the trig equations or SOH CAH TOA to get "h", the perpendicular line.

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

   This relates to the area formula that we are familiar with because you are simply using sin, cos, or tan to find "h" which is the perpendicular line that we "dropped" since these triangles are not right triangles. The area of an oblique triangle needs to contain two sides and an included angle. Therefore, it has to be SAS in order to use the area of an oblique triangle.

References:
Mrs. Kirch's SSS packet