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6.6 Summary (EMBHT)

square identity

quotient identity

\(\cos^2\theta + \sin^2\theta = 1\)\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
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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 \(\triangle ABC = \frac{1}{2} ab \sin C\)\(c^2 = a^2 + b^2 - 2 ab \cos C\)
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General solution:

  1. \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*}
  2. \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*}
  3. \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