\(\begin{aligned} \dfrac{a}{\sin A}=&\dfrac{b}{\sin B}=\dfrac{c}{\sin C} \\\\ &\quad \text{or} \\\\ \dfrac{\sin A}{a}=&\dfrac{\sin B}{b}=\dfrac{\sin C}{c} \end{aligned}\)
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Example:
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In the diagram, \(LMN\) is a triangle.
Find the length of \(LN\).
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\(\begin{aligned} \angle{M}&=180^\circ -85^ \circ-40^\circ \\\\ &=55 ^\circ. \end{aligned}\)
\(\begin{aligned} \dfrac{LN}{\sin 55^\circ}&=\dfrac{7}{ \sin 40 ^\circ} \\\\ LN&= \dfrac{7}{ \sin 40 ^\circ} \times \sin 55^\circ \\\\ &= 8.92 \text{ cm}. \end{aligned}\)
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| \(\blacksquare\) Sine rule can be used to find the unknown angles or sides of a triangle when |
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- Two angles and one side are known.
- Two sides and one non-included angle are known.
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\(\blacksquare\) Ambiguous case
When two sides and one non-included angle \((a,\,b,\,\angle A)\) are given, an ambiguous case occurs if
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- \(a \lt b\)
- \(\angle A\) is an acute angle
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| Two triangles exist \((AB_1C, AB_2C)\). |
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Example:
Given a triangle \(ABC\) such that
\(\begin{aligned} AB&=6.2 \text{ cm}, \\\\ AC &=4.8 \text{ cm}, \\\\ &\text{ and} \\\\ \angle ABC&=43 ^\circ. \end{aligned}\)
Find the possible values of \(\begin{aligned} \angle BCA \end{aligned}\).
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\(\begin{aligned} \dfrac{\sin C}{6.2}&=\dfrac{\sin 43^\circ}{ 4.8} \\\\ \angle BC_1A&=\sin^{-1} \left( \dfrac{\sin 43^\circ}{4.8} \times 6.2 \right) \\\\ &=61.75 ^ \circ \\\\ \angle BC_2A&= 180 ^\circ-61.75 ^ \circ \\\\ &=118.25 ^\circ \end{aligned}\)
