Sketching Tangent Graphs

Sketch the graph of \(f(\theta) = \tan \cfrac{1}{2}(\theta - \text{30}\text{°})\) for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\).

Examine the form of the equation

From the equation we see that \(0 < k < 1\), therefore the branches of the graph will be less steep than the standard tangent graph \(y = \tan \theta\). We also notice that \(p < 0\) so the graph will be shifted to the right on the \(x\)-axis.

Determine the period

The period for \(f(\theta) = \tan \cfrac{1}{2}(\theta - \text{30}\text{°})\) is:

\begin{align*} \text{Period} &= \cfrac{\text{180}\text{°}}{|k|} \\ &= \dfrac{\text{180}\text{°}}{\cfrac{1}{2}} \\ &= \text{360}\text{°} \end{align*}

Determine the asymptotes

The standard tangent graph, \(y = \tan \theta\), for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\) is undefined at \(\theta = -\text{90}\text{°}\) and \(\theta = \text{90}\text{°}\). Therefore we can determine the asymptotes of \(f(\theta) = \tan \cfrac{1}{2}(\theta - \text{30}\text{°})\):

• \(\cfrac{-\text{90}\text{°}}{\text{0.5}} + \text{30}\text{°} = -\text{150}\text{°}\)
• \(\cfrac{\text{90}\text{°}}{\text{0.5}} + \text{30}\text{°} = \text{210}\text{°}\)

The asymptote at \(\theta = \text{210}\text{°}\) lies outside the required interval.

Plot the points and join with a smooth curve

Period: \(\text{360}\text{°}\)

Domain: \(\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{150}\text{°} \}\)

Range: \((-\infty;\infty)\)

\(y\)-intercepts: \((\text{0}\text{°};-\text{0.27})\)

\(x\)-intercept: \((\text{30}\text{°};0)\)

Asymptotes: \(\theta = -\text{150}\text{°}\)