Mathematics » Functions II » The Sine Function

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

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

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 $$(\text{160}\text{°}; 0)$$