We will prove the Law of Sines by filling in the blanks for the given column proof. Let's first recall what this law says.
To prove this law, we begin by drawing the altitude CD of â–ł ABC.
By the definition of altitude, we have that both â–ł ACD and â–ł CBD are right triangles. These are the first two steps written in the given table. Now, we will continue by filling in the blanks.
Filling in the Blanks
Since â–ł ACD and â–ł CBD are right triangles, we can use the definition of sine to write the two relations below.
3)& sin A = h/b, sin B = h/a
3)& a. definition of sine
Next, we solve each equation for h.
4)& bsin A = h, asin B = h
4)& b. solving for h
By substituting the second equation into the first one, we get the equation below.
5)& bsin A = asin B
5)& c. substitution
Finally, we divide each side by a b and simplify to get the desired result.
6)& sin A/a = sin B/b
6)& d. dividing by a b
