We think you are located in South Africa. Is this correct?

Summary

Test yourself now

High marks in maths are the key to your success and future plans. Test yourself and learn more on Siyavula Practice.

Sign up and test yourself

6.6 Summary (EMBHT)

square identity

quotient identity

\(\cos^2\theta + \sin^2\theta = 1\)\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
41e09cda20df9b26b594f4f83f0b47ae.png

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 \(\triangle ABC = \frac{1}{2} ab \sin C\)\(c^2 = a^2 + b^2 - 2 ab \cos C\)
bfe67719a672d19c3cd42077c2e84ea6.png

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