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Frequently Asked Questions

Can you prove the general triangle trigonmetric formulae?


Consider the diagram.

In the left hand triangle, x = b sin A, and in the right hand triangle, x = a sin B, therefore b sin A = a sin B

So we get,
sin A

a
=
sin B

b
. This result is called the sine rule.

By combining right-angle trigonometry and the Pythagorean Theorem we can establish the cosine rule. Which states that in the general triangle,
c² = a² + b² – 2ab cosθ.

It would make sense that c² = a² + b² – ?. In other words, we use the Pythagorean Theorem with some adjustment that takes not being a right angle into account.

Applying the Pythagorean Theorem to the left hand triangle, a² = x² + h².

In the right hand triangle, c² = (b – x)² + h² = b² – 2bx + x² + h².

As a² = x² + h² we get c² = b² – 2bx + a².

But x = a cosθ, hence c² = b² – 2b × a cosθ + a².

This can be written as c² = a² + b² – 2ab cosθ, which is as we expected; the Pythagorean Theorem with an adjustment.