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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 β2β and dividing by βabc:
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 ambiguous case of the sine rule is a consequence of the symmetry of the βsinβ function:
Therefore, if a given triangle has a specific βsin(A)β and it is not specified whether βAβ is acute / obtuse, then there are two possible lengths for the side opposite βA:
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When we type βsinβ1(x)β into the calculator, it will always return an acute angle. To find the corresponding obtuse angle, we take
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.