Mathematics » Functions II » The Tangent Function

Revision of The Tangent Function

Revision of The Tangent Function

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

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The dashed vertical lines are called the asymptotes. The asymptotes are at the values of θ where \(\tan\theta\) is not defined.

  • Period: \(\text{180}\text{°}\)

  • Domain: \(\{\theta : \text{0}\text{°} \le \theta \le \text{360}\text{°}, \theta \ne \text{90}\text{°}; \text{270}\text{°}\}\)

  • Range: \(\{f(\theta):f(\theta)\in ℝ\}\)

  • \(x\)-intercepts: \((\text{0}\text{°};0)\), \((\text{180}\text{°};0)\), \((\text{360}\text{°};0)\)

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

  • Asymptotes: the lines \(\theta =\text{90}\text{°}\) and \(\theta =\text{270}\text{°}\)

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

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

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

  • 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 \(a>1\), branches of \(f(\theta)\) are steeper.

    • For \(0<a<1\), branches of \(f(\theta)\) are less steep and curve more.

    • 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 branches of the graph are less steep.

    • For \(a < -1\), there is a reflection about the \(x\)-axis and the branches of the graph are steeper.

 

\(a<0\)

\(a>0\)

\(q>0\)

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\(q=0\)

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\(q<0\)

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