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The sine law can be derived using a non-right triangle.


First, split the triangle into two right triangles by drawing an altitude from vertex \(B\), intersecting side \(AC\) at a right angle. Label this height \(h_1\).

 

In the Investigation, you saw that expressing \(\color{blue}{h_1}\) in terms of each hypotenuse and the sine of the angle opposite \(\color{blue}{h_1}\) resulted in a relationship between side \(a\), angle \(A\), side \(c\), and angle \(C\).

\[\begin{align}
 \sin A &= \frac{{\color{blue}{h_1} }}{c} \\
 c\sin A &= \color{blue}{h_1} \\
 \end{align}\]
\[\begin{align}
 \sin C &= \frac{{\color{blue}{h_1} }}{a} \\
a\sin C &= \color{blue}{h_1} \\
 \end{align}\]

Equating the two equations:
\[\begin{align}
 a\sin C &= c\sin A \\
 \frac{a}{{\sin A}} &= \frac{c}{{\sin C}} \\
 \end{align}\]

In order to determine a relationship that includes side \(b\) and angle \(B\), split the triangle into two different right triangles by drawing an altitude from vertex \(C\), intersecting side \(AB\) at a right angle. Label this height \(\color{green}{h_2}\).



Similar to the process for \(h_1\), use the upper and lower triangles to express \(\color{green}{h_2}\) in terms of each hypotenuse and the sine of the angle opposite \(\color{green}{h_2}\).
\[\begin{align}
 \sin B &= \frac{{\color{green}{h_2} }}{a} \\
 a\sin B &= \color{green}{h_2} \\
 \end{align}\]
\[\begin{align}
 \sin A &= \frac{{\color{green}{h_2} }}{b} \\
 b\sin A &= \color{green}{h_2} \\
 \end{align}\]

Equate the two equations:
\[\begin{align}
 a\sin B &= b\sin A \\
 \frac{a}{{\sin A}} &= \frac{b}{{\sin B}} \\
 \end{align}\]

Since \(\frac{a}{{\sin A}} = \frac{c}{{\sin C}}\) and \(\frac{a}{{\sin A}} = \frac{b}{{\sin B}}\), it follows that \(\frac{b}{{\sin B}} = \frac{c}{{\sin C}}\).

Therefore, \(\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}\). This relationship relating the sides and angles of a non-right triangle is called the sine law.