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.