Mathematics » Functions II » The Cosine Function

Revision of The Cosine Function

Functions of the form \(y = \cos \theta\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

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  • The period is \(\text{360}\text{°}\) and the amplitude is \(\text{1}\).

  • Domain: \([\text{0}\text{°};\text{360}\text{°}]\)

    For \(y = \cos \theta\), the domain is \(\{ \theta: \theta \in \mathbb{R} \}\), however in this case, the domain has been restricted to the interval \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

  • Range: \([-1;1]\)

  • \(x\)-intercepts: \((\text{90}\text{°};0)\), \((\text{270}\text{°};0)\)

  • \(y\)-intercept: \((\text{0}\text{°};1)\)

  • Maximum turning points: \((\text{0}\text{°};1)\), \((\text{360}\text{°};1)\)

  • Minimum turning point: \((\text{180}\text{°};-1)\)

Functions of the form \(y = a \cos \theta + q\)

Cosine functions of the general form \(y = a \cos \theta + q\), where \(a\) and \(q\) are constants.

The effects of \(a\) and \(q\) on \(f(\theta) = a \cos \theta + q\):

  • The effect of \(q\) on vertical shift

    • For \(q>0\), \(f(\theta)\) is shifted vertically upwards by \(q\) units.

    • For \(q<0\), \(f(\theta)\) is shifted vertically downwards by \(q\) units.

  • The effect of \(a\) on shape

    • For \(a>1\), the amplitude of \(f(\theta)\) increases.

    • For \(0<a<1\), the amplitude of \(f(\theta)\) decreases.

    • For \(a<0\), there is a reflection about the \(x\)-axis.

    • For \(-1 < a < 0\), there is a reflection about the \(x\)-axis and the amplitude decreases.

    • For \(a < -1\), there is a reflection about the \(x\)-axis and the amplitude increases.

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