The sine law of triangles can be proved by dividing the triangle into right-triangles.
Option (c) proves the first equality in the law of sines
From the question, we have:
h equals to the distance from point C to segment AB
The sine of angle is:
[tex]\mathbf{sin(\theta) = \frac{Opposite}{Hypotenuse}}[/tex]
For angle at A, we have:
[tex]\mathbe{\theta = A}[/tex]
[tex]\mathbf{Hypotenuse = b}\\\mathbf{Opposite = h}[/tex]
So, we have:
[tex]\mathbf{sin(\theta) = \frac{Opposite}{Hypotenuse}}[/tex]
Substitute values for Opposite, Hypotenuse and theta
[tex]\mathbf{sin A = \frac{h}{b}}[/tex]
Similarly
[tex]\mathbf{sin B = \frac{h}{a}}[/tex]
Make h the subject
[tex]\mathbf{h = a\ sinB}[/tex]
Substitute [tex]\mathbf{h = a\ sinB}[/tex] in [tex]\mathbf{sin A = \frac{h}{b}}[/tex]
[tex]\mathbf{sin A = \frac{a\ sinB}{b}}[/tex]
Divide both sides by a
[tex]\mathbf{\frac{sin A}{a} = \frac{sinB}{b}}[/tex]
Hence, the correct option is (c).
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