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Non-right-angled triangles
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Sine and cosine rules, finding general area of a triangle
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Area of non-right-angled triangles
SL Core 3.2
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
sinC=ah⇒h=asinC
Thus
A=21(bh)=21absinC📖
Sine rule
SL Core 3.2
The previously found formula for area
A=21absinC📖
applies to any pair of sides and the angle between them. Since the area is the same no matter which sides we use:
21absinC=21bcsinA=21acsinB
Multiplying everything by 2 and dividing by abc:
csinC=asinA=BsinB🚫
Flipping the numerator and denominator gives the form that appears in the formula booklet:
sinAa=sinBb=sinCc📖
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.
Ambiguous Case of the Sine rule
SL 3.5
The ambiguous case of the sine rule is a consequence of the symmetry of the sin function:
sin(180°−A)=sin(A)
Therefore, if a given triangle has a specific sin(A) and it is not specified whether A is acute / obtuse, then there are two possible lengths for the side opposite A:
When we type sin−1(x) into the calculator, it will always return an acute angle. To find the corresponding obtuse angle, we take
180°−sin−1(x)
Cosine rule
SL Core 3.2
The cosine rule is a generalization of Pythagoras' theorem for non-right-angled triangles. It states that
c2=a2+b2−2abcosC📖
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.
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