Question

a d c b bret sin(c) = c/h sin(a) = a/h h = a/sin(a) h = c/sin(c) a/sin(a) = c/sin(c) sin(c)/c = sin(a)/a kamala sin(c) = h/a sin(a) = h/c asin(c) = h csin(a) = h asin(c) = csin(a) sin(c)/c = sin(a)/a the proof was correctly completed by the next step in the proof is to and write an expression for both bret and kamala kamala bret reset next

Explanation:

Step1: Recall sine - rule concept

The sine - rule in a triangle states that $\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$, where $a$, $b$, $c$ are the side - lengths of a triangle and $A$, $B$, $C$ are the opposite angles respectively.

Step2: Analyze Bret's work

Bret starts with the definitions of sine in right - triangles formed within the larger triangle. He uses $\sin(C)=\frac{c}{h}$ and $\sin(A)=\frac{a}{h}$, then solves for $h$ in two ways ($h = \frac{a}{\sin(A)}$ and $h=\frac{c}{\sin(C)}$) and equates the two expressions for $h$ to get $\frac{a}{\sin(A)}=\frac{c}{\sin(C)}$ and then $\frac{\sin(C)}{c}=\frac{\sin(A)}{a}$.

Step3: Analyze Kamala's work

Kamala also starts with the sine definitions in right - triangles. She has $\sin(C)=\frac{h}{a}$ and $\sin(A)=\frac{h}{c}$, then cross - multiplies to get $a\sin(C)=h$ and $c\sin(A)=h$, and then equates the two expressions for $h$ ($a\sin(C)=c\sin(A)$) and finally gets $\frac{\sin(C)}{c}=\frac{\sin(A)}{a}$.

Answer:

The proof was correctly completed by both Bret and Kamala.