6.6 Summary | Trigonometry

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	co-function 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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