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Imagine a random triangle ABC. It will make it easier if you draw it with angle C at the top. The opposite side of angle A is labelled a, the opposite side of angle B is labelled b and the opposite side of angle C is labelled c. Draw the altitude (height) from angle C so that it is perpendicular (at 90 degrees) to side c.

Looking at this triangle, find expressions for the sines of angles A and B:

sinA = h/b

sinB = h/a

Rearrange these two equations in terms of h:

h = bsinA

h = asinB

As h = h, these equations can be set equal to each other and simplified to find the sine rule:

bsinA = asinB

sinA/a = sinB/b

If you expand on this way of working, you can also find that sinA/a = sinB/b = sinC/c. You have now proven the sine rule for all triangles!

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12y ago

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Q: How do you prove the sine rule?
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