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Non-right-angled triangles
Sine and cosine rules, finding general area of a triangle
Sine and cosine rules, finding general area of a triangle
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When we know two side lengths and the angle between the two sides, we can find the area even if we don't directly know the height by using the fact that
Thus
The previously found formula for area
applies to any pair of sides and the angle between them. Since the area is the same no matter which sides we use:
Multiplying everything by ​2​ and dividing by ​abc:
Flipping the numerator and denominator gives the form that appears in the formula booklet:
The sine rule is primarily used when we know two angles and a side. When we know two sides and an angle, the version with angles in the numerator is easier to work with.
The cosine rule is a generalization of Pythagoras' theorem for non-right-angled triangles. It states that
The cosine rule is primarily used when we
know two sides and the angle between them, and want to find the third side,
know all three sides and want to find an angle.