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

   1. Law of Sines:
                                                                           
                                         

   Why do we need it?

       We need the law of sines because not every triangle is a right triangle, so we use the law of sines to solve for non-right triangles. We cannot use the Pythagorean Theorem or the normal trig function to solve for non-right triangles like we do for right triangles, but the law of sines luckily works for any triangle.

     How is it derived from what we already know?

·         The first thing we do is draw and label our non-right triangle.

·         We then drop down a perpendicular from “angle B,” which makes two triangles, and we call it “h”.

·         Since we now have two triangles, we can use the normal trig functions like sin, cos, tan, csc, sec, and cot; so “we can drop perpendiculars from the other two vertices and get the other relationships.”

·         After getting the two sin ratios, we see that they have “h” in common, so after we simplify both of the equations by getting rid of the denominators, we equal both to each other since both of them equal to “h”. We then get cSinA=aSinC.

·         We then want to get Sin and its angle by itself so we divide by the coefficients, which are “a” and “c” in this case, and you then get “sinA/a = sinC/c”.

·         It all depends where you make your perpendicular line “h”. So we can find angle B if only the perpendicular line is from a different angle like A or C and you do the same process.

 4. Area of an Oblique Triangle:

1     Area formulas:

·   “The area of a triangle is A=1/2bh, where ‘b’ is the base and ‘h’ is the perpendicular height of the triangle”.

·    “The area of an oblique (all sides’ different lengths) triangle is one-half of the product of two sides and the sine of their included angle (the angle in between them)”.

·    Formula: It depends on what angles they give you, but the options you have are A=1/2bcsinA, A= 1/2acsinB, or A= 1/2absinC.

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

·    The area of an oblique triangle is derived by cutting the triangle into two, which makes the triangle into a right triangle; this is done by using “h” as a line. Because we now have two right triangles, we can use the normal trig functions. Depending where “h” is dropped down from a different angle, we can see which one we are using from the options of sinC=h/a, sinA=h/c, sinB=h/(a or c). Angle B has two options because “h” is dropped down so it can apply to both triangles.

·    Since we want to get “h” by itself, we multiply by the denominator to both sides to get rid of it.

·    We then plug each of the equations into the formula for the area of a triangle, which is “A=1/2bh”.

·    We then substitute in our regular area equation for “h,” and we get A=1/2b(aSinC) or A=1/2b(cSinA). This also applies to angle B when you do it in the other triangle.

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

It relates to the area formula that we are familiar with by plugging in the values into “h,” which make new formulas, and it also uses the normal trig formula for the area of a triangle. The area of an oblique is conditional; it needs to obtain two side lengths, which cannot be the same letter, and their angles must be included. Or else it will not work.

References:

Unit P SSS Packet