6.6 Summary
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6.5 Area, sine, and cosine rules

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6.6 Summary (EMBHT)
square identity 
quotient identity 
\(\cos^2\theta + \sin^2\theta = 1\)  \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\) 
negative angles 
periodicity identities 
cofunction identities 
\(\sin (\theta) =  \sin \theta\)  \(\sin (\theta \pm \text{360}\text{°}) = \sin \theta\)  \(\sin (\text{90}\text{°}  \theta) = \cos \theta\) 
\(\cos (\theta) = \cos \theta\)  \(\cos (\theta \pm \text{360}\text{°}) = \cos \theta\)  \(\cos (\text{90}\text{°}  \theta) = \sin \theta\) 
sine rule 
area rule 
cosine rule 
\(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)  area \(\triangle ABC = \frac{1}{2} bc \sin A\)  \(a^2 = b^2 + c^2  2 bc \cos A\) 
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)  area \(\triangle ABC = \frac{1}{2} ac \sin B\)  \(b^2 = a^2 + c^2  2 ac \cos B\) 
area \(\triangle ABC = \frac{1}{2} ab \sin C\)  \(c^2 = a^2 + b^2  2 ab \cos C\) 
General solution:
 \begin{align*} \text{If } \sin \theta &= x \\ \theta &= \sin^{1}x + k \cdot \text{360}\text{°} \\ \text{or } \theta &= \left( \text{180}\text{°}  \sin^{1}x \right) + k \cdot \text{360}\text{°} \end{align*}
 \begin{align*} \text{If } \cos \theta &= x \\ \theta &= \cos^{1}x + k \cdot \text{360}\text{°} \\ \text{or } \theta &= \left( \text{360}\text{°}  \cos^{1}x \right) + k \cdot \text{360}\text{°} \end{align*}
 \begin{align*}
\text{If } \tan \theta &= x \\
\theta &= \tan^{1}x + k \cdot \text{180}\text{°}
\end{align*}
for \(k \in \mathbb{Z}\).
How to determine which rule to use:

Area rule:
 no perpendicular height is given

Sine rule:
 no right angle is given
 two sides and an angle are given (not the included angle)
 two angles and a side are given

Cosine rule:
 no right angle is given
 two sides and the included angle angle are given
 three sides are given
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6.5 Area, sine, and cosine rules

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