BQ#1: Unit P Concept 1 and 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 solve triangles that are not right triangles. The normal trig functions are defined for a right triangle and are not directly useful in solving non right triangles but we can use the trig function to determine the law of sines. So first you have a non right triangles labeled as ABC. Then you make a imaginary perpendicular line from angle B and we call that h. Now we have formed two triangles. Now we use the trig function of sin which is hypotenuse over adjacent and then on the other triangle we have the same as well but we don't know the numbers so for the triangle on the left side it is SinA=h/c and for the right it is SinC=h/a. We then want h by itself so we multiply by c and on the other by a and we get cSinA=h and the other aSinC=h then we equal them together cSinA=aSinC and you divide by ac and you get SinA/a=SinC/c and that is exactly what the law of sines is.

To get a picture in your head of how to do it instead of me telling you there is a video here.

4.) Area Formula- How is the "area of an oblique" triangle derived? How does it relate to the area formula that you are familiar with? 
       The area formula is derived from the area equation and the trig function. The area equation you should know form Geometry is A=1/2bh where b is the base and h is the height. We then draw a perpendicular line on triangle ABC in half from angle B to create two triangles with a sharing side of h. We then know that SinC=h/a and when you get h by itself you get h=aSinC. Then you substitute it in the area equation for h and we get A=1/2b(aSinC). The area of an oblique triangle is one half of the product of two sides and the Sine of their included angle.

To get a visual in your head of how it should look like and example problems as well watch this video here.
      It relates to the area formula by using 1/2bh as a start to find the area formula for an oblique triangle. We still use 1/2 and it still consists of two sides and the base for b but we have to find our own h by using the trig function of sine.

Works Cited
http://www.youtube.com/watch?v=zJsw_ltZxFo
http://www.youtube.com/watch?v=gAX_IleqeJQ