Mathematics » Functions II » The Sine Function

Sketching Sine Graphs

Sketching Sine Graphs

Example

Question

Sketch the graph of \(f(\theta) = \sin (\text{45}\text{°} – \theta)\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

Examine the form of the equation

Write the equation in the form \(y = \sin (\theta + p)\).

\begin{align*} f(\theta) &= \sin (\text{45}\text{°} – \theta)\\ &= \sin (-\theta + \text{45}\text{°}) \\ &= \sin ( -(\theta – \text{45}\text{°}) ) \\ &= -\sin (\theta – \text{45}\text{°}) \end{align*}

To draw a graph of the above function, we know that the standard sine graph, \(y = \sin\theta\), must:

  • be reflected about the \(x\)-axis
  • be shifted to the right by \(\text{45}\text{°}\)

Complete a table of values

θ\(\text{0}\)\(\text{°}\)\(\text{45}\)\(\text{°}\)\(\text{90}\)\(\text{°}\)\(\text{135}\)\(\text{°}\)\(\text{180}\)\(\text{°}\)\(\text{225}\)\(\text{°}\)\(\text{270}\)\(\text{°}\)\(\text{315}\)\(\text{°}\)\(\text{360}\)\(\text{°}\)
\(f(\theta)\)\(\text{0.71}\)\(\text{0}\)\(-\text{0.71}\)\(-\text{1}\)\(-\text{0.71}\)\(\text{0}\)\(\text{0.71}\)\(\text{1}\)\(\text{0.71}\)

Plot the points and join with a smooth curve

Sketching Sine Graphs

Period: \(\text{360}\text{°}\)

Amplitude: \(\text{1}\)

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

Range: \([-1;1]\)

Maximum turning point: \((\text{315}\text{°};1)\)

Minimum turning point: \((\text{135}\text{°};-1)\)

\(y\)-intercepts: \((\text{0}\text{°};\text{0.71})\)

\(x\)-intercept: \((\text{45}\text{°};0) \text{ and } (\text{225}\text{°};0)\)

Example

Question

Sketch the graph of \(f(\theta) = \sin (3\theta + \text{60}\text{°})\) for \(\text{0}\text{°} \leq \theta \leq \text{180}\text{°}\).

Examine the form of the equation

Write the equation in the form \(y = \sin k(\theta + p)\).

\begin{align*} f(\theta) &= \sin (3\theta + \text{60}\text{°})\\ &= \sin 3(\theta + \text{20}\text{°}) \end{align*}

To draw a graph of the above equation, the standard sine graph, \(y = \sin\theta\), must be changed in the following ways:

  • decrease the period by a factor of \(\text{3}\);
  • shift to the left by \(\text{20}\text{°}\).

Complete a table of values

θ\(\text{0}\)\(\text{°}\)\(\text{30}\)\(\text{°}\)\(\text{60}\)\(\text{°}\)\(\text{90}\)\(\text{°}\)\(\text{120}\)\(\text{°}\)\(\text{150}\)\(\text{°}\)\(\text{180}\)\(\text{°}\)
\(f(\theta)\)\(\text{0.87}\)\(\text{0.5}\)\(-\text{0.87}\)\(-\text{0.5}\)\(\text{0.87}\)\(\text{0.5}\)\(-\text{0.87}\)

Plot the points and join with a smooth curve

Sketching Sine Graphs

Period: \(\text{120}\text{°}\)

Amplitude: \(\text{1}\)

Domain: \([\text{0}\text{°}; \text{180}\text{°}]\)

Range: \([-1;1]\)

Maximum turning point: \((\text{10}\text{°}; 1) \text{ and } (\text{130}\text{°}; 1)\)

Minimum turning point: \((\text{70}\text{°}; -1)\)

\(y\)-intercept: \((\text{0}\text{°}; \text{0.87})\)

\(x\)-intercepts: \((\text{40}\text{°}; 0)\), \((\text{100}\text{°}; 0)\) and \((\text{160}\text{°}; 0)\)

Do you want to suggest a correction or an addition to this content? Leave Contribution

[Attributions and Licenses]


This is a lesson from the tutorial, Functions II and you are encouraged to log in or register, so that you can track your progress.

Log In

Share Thoughts


Do NOT follow this link or you will be banned from the site!