Question

question 5
5.1 using the diagram below, prove that $\frac{sin a}{a}=\frac{sin b}{b}$

Explanation:

Step1: Draw an altitude

Draw the altitude \(h\) from vertex \(C\) to side \(AB\). Let the foot of the altitude be \(D\).

Step2: Express \(\sin A\) and \(\sin B\)

In right - triangle \(ACD\), \(\sin A=\frac{h}{b}\), so \(h = b\sin A\). In right - triangle \(BCD\), \(\sin B=\frac{h}{a}\), so \(h=a\sin B\).

Step3: Equate the two expressions for \(h\)

Since \(h = b\sin A\) and \(h=a\sin B\), we have \(b\sin A=a\sin B\).

Step4: Rearrange the equation

Dividing both sides of the equation \(b\sin A=a\sin B\) by \(ab\), we get \(\frac{\sin A}{a}=\frac{\sin B}{b}\).

Answer:

The proof is completed as above, showing that \(\frac{\sin A}{a}=\frac{\sin B}{b}\).