Question

bret
\\( \sin ( c ) = \frac { c } { h } \\)
\\( \sin ( a ) = \frac { a } { h } \\)
\\( h = \frac { a } { \sin ( a ) } \\)
\\( h = \frac { c } { \sin ( c ) } \\)
\\( \frac { a } { \sin ( a ) } = \frac { c } { \sin ( c ) } \\)
\\( \frac { \sin ( c ) } { c } = \frac { \sin ( a ) } { a } \\)
kamala
\\( \sin ( c ) = \frac { h } { a } \\)
\\( \sin ( a ) = \frac { h } { c } \\)
\\( a \sin ( c ) = h \\)
\\( c \sin ( a ) = h \\)
\\( a \sin ( c ) = c \sin ( a ) \\)
\\( \frac { \sin ( c ) } { c } = \frac { \sin ( a ) } { a } \\)
the proof was correctly completed by
the next step in the proof is to and write an expression for

Explanation:

Step1: Analyze Bret's work

Bret's first two equations \( \sin(C)=\frac{c}{h}\) and \( \sin(A)=\frac{a}{h}\) are incorrect. In a right - triangle, the sine of an angle is defined as \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle \(C\) in the right - triangle with height \(h\), \( \sin(C)=\frac{h}{a}\) (opposite side is \(h\) and hypotenuse is \(a\)), and for angle \(A\), \( \sin(A)=\frac{h}{c}\) (opposite side is \(h\) and hypotenuse is \(c\)).

Step2: Analyze Kamala's work

Kamala correctly uses the sine formula \( \sin(C)=\frac{h}{a}\Rightarrow a\sin(C) = h\) and \( \sin(A)=\frac{h}{c}\Rightarrow c\sin(A)=h\). Then by equating \(a\sin(C)\) and \(c\sin(A)\) (since both equal \(h\)), we get \(a\sin(C)=c\sin(A)\), and further \( \frac{\sin(C)}{c}=\frac{\sin(A)}{a}\).

Step3: Consider the next step in the Law of Sines proof

The Law of Sines states that \( \frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\). To complete the proof, we need to draw a new perpendicular line (from vertex \(B\) to side \(AC\) if we haven't already considered all angles) and write an expression for the sine of \(B\). Using the same right - triangle definition of sine (\( \sin(B)=\frac{h_{new}}{c}\) or \( \sin(B)=\frac{h_{new}}{a}\) depending on the perpendicular drawn), and then relating it to the other sides.

Answer:

The next step in the proof is to draw in a new perpendicular line and write an expression for the sine of \(B\).