BQ #1: Unit P Concepts 6-7 (Law of Sines and Law of Cosines)

Law of Sines

1. Why do we need it?

• First and foremost, we have to understand that not all triangles are right triangles. This means that not all triangles have a ninety degree angle, meaning we cannot use the Pythagorean theorem to solve for more. Then how will we solve  for irregular triangles you ask? Fear not, my beloved friend because this is where the Law of Sines come in to play.

-The Law of Sines, however, may only be used when given SSA (side-side-angle)

2. How do we derive from what we already know?

• From what we already know, the triangle does not have a right triangle. this is where we draw a Line straight down the middle to make two right triangles. This length is labeled as "h."

 

http://www.regentsprep.org/Regents/math/algtrig/ATT12/derivelawofsines.htm

 

• If we look at this from a graph perspective, we see that the sin<A will equal h/c because it's opposite over adjacent. If we multiply both sides by "c" then we would get cSin<A=h. Sin<C will equal h/a and multiply by "a" then aSin<C will also equal "h" which will make all angles of the same value.
• This can also be said for angle B therefore we plug Sin<A/value = Sin<B/x and cross multiply! This is how we derive the Law of Sine.

 

Area of an Oblique Triangle 

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

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

 

• The Area of an Oblique Triangle is derived by creating our own "base and height" on the actual triangle. As shown in figure-left, we had to drop the perpendicular line straight down to get our height but as seen in figure-right we had to make the height and the added base. Remember that we are still using our trig functions. The formula to find the area of an Oblique triangle:http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg
•  
• Because there are different angles, there will be different variations of the equation. What we must be given is a Side-Angle-Side.

 

2. How does this relate to the formula we already know?

This relates to the formula because we are still using our regular trig functions: sin, cos, and tan. We use the original formula of H=1/2(bh) as the origin of the triangle and depending on what triangle we are given, we use the same height and base.