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Non-right-angled triangles
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Area of non-right-angled triangles usingΒ βA=21βabsinC,Β the sine ruleΒ β,Β
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
β
sinC=ahββh=asinC
βThus
β
A=21β(bh)=21βabsinCπ
βSine rule
SL Core 3.2
The previously found formula for area
β
A=21βabsinCπ
βapplies to any pair of sides and the angle between them. Since the area is the same no matter which sides we use:
β
21βabsinC=21βbcsinA=21βacsinB
βMultiplying everything byΒ β2βΒ and dividing byΒ βabc:
β
csinCβ=asinAβ=BsinBβπ«
βFlipping the numerator and denominator gives the form that appears in the formula booklet:
β
sinAaβ=sinBbβ=sinCcβπ
βThe 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 theΒ βsinβΒ function:
β
sin(180Β°βA)=sin(A)
βTherefore, if a given triangle has a specificΒ βsin(A)βΒ and it is not specified whetherΒ βAβΒ is acute / obtuse, then there are two possible lengths for the side oppositeΒ βA:
When we typeΒ βsinβ1(x)βΒ into the calculator, it will always return an acute angle. To find the corresponding obtuse angle, we take
β
180Β°βsinβ1(x)
βCosine rule
SL Core 3.2
The cosine rule is a generalization of Pythagoras' theorem for non-right-angled triangles. It states that
β
c2=a2+b2β2abcosCπ
βThe 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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