Circular functions graphs

Sine, cosine and tangent functions are periodic functions, meaning they repeat themselves at regular intervals. This interval of repetition is known as the period of the function, and this periodicity leads to many of the features of the graphs of these functions. Recognizing the periodic nature of these functions facilitates the predicting and analyzing of patterns in a wide range of applications, from physics and engineering to signal processing and music. It also simplifies solving equations involving trigonometric functions by identifying recurring solutions.

Features of Sine, Cosine and Tangent Function Graphs

Sin and Cos

Sine and cosine functions show periodicity and symmetry. Graphs of sine and cosine functions further share similar features including a similar shape, period, amplitude and mean (or midline) position.

Features of \(y=\sin(x)\)

  • A repeating wave pattern, with a period of \(2\pi\)
  • Oscillates between \(y=-1\) and \(y=1\), with a mean value, or midline, of \(y=0\).
  • Waves have an amplitude (distance from mean to max value) of \(1\).
  • \(x\)-intercepts at \(x=k\pi\) where \(k \in \mathbb{Z}\).
  • \(y\)-intercept at the origin \((0, 0)\).

A graph of the sine function, y = sin(x), showing its key properties. The curve oscillates between -1 and 1, with a maximum at y = 1 and a minimum at y = -1. The amplitude is marked as 1 (magenta), the mean value is y = 0 (green), and the period is labelled as 2π (orange). The function has a repeating wave pattern with intercepts at integer multiples of π. The graph highlights the fundamental characteristics of the sine function, including periodicity and symmetry

Features of \(y=\cos(x)\)

  • A repeating wave pattern, with a period of \(2\pi\)
  • Oscillates between \(y=-1\) and \(y=1\), with a mean value, or midline, of \(y=0\).
  • Waves have an amplitude (distance from mean to max value) of \(1\).
  • \(x\)-intercepts at \(x=\frac{\pi}{2}+k\pi\) where \(k \in \mathbb{Z}\).
  • \(y\)-intercept at \((1, 0)\).

A graph of the cosine function, y = cos(x), showing its key properties. The curve oscillates between -1 and 1, with a maximum at y = 1 and a minimum at y = -1. The amplitude is marked as 1 (magenta), the mean value is y = 0 (green), and the period is labelled as 2π (orange). The function has a repeating wave pattern with intercepts at odd integer multiples of π/2. The graph highlights the fundamental characteristics of the cosine function, including periodicity and symmetry.

Common Features

  • Domain: \(x \in \mathbb{R}\)
  • Range:  \(y \in [-1, 1]\)
  • Mean: \(y = 0\)
  • Period \(=2\pi\)
  • Amplitude \(=1\)

The graph of \(\cos(x)\) is equivalent to a graph of \(\sin(x)\) that has been shifted \(\frac{\pi}{2}\) units in the negative direction of the \(x\)-axis.

Tan

While graphs for cosine and sine functions are similar, those for tangent functions differ significantly from them, sharing only in that they show periodicity and symmetry. Recall
\(\tan(x)= \dfrac{\sin(x)}{\cos(x)}\).

Therefore, when \(\cos(x) = 0 ,\ \tan(x)\) is undefined and will have vertical asymptotes at these values for \(x\). Further, while sin and cos functions have a strict minimum and maximum, tan function extend for all real values of \(y\).

Features of \(y=\tan(x)\)

  • Maximum value of \(y = \infty\), minimum of \(y= -\infty\)
  • Mean of \(y = 0\)
  • Period of \(\pi\) units
  • Asymptotes (values that aren’t defined) at \(x=\frac{\pi}{2}+k\pi\), where  \(k \in \mathbb{Z}\).
  • \(x\)-intercepts at \(x=k\pi\), where \(k \in \mathbb{Z}\).

A graph of the tangent function, y = tan(x), showing its key properties. The function has a period of 2π (labelled in orange) and vertical asymptotes at odd multiples of π/2, where the function is undefined. The curve passes through the origin (0,0) and exhibits repeating steep, S-shaped segments. The function increases without bound as it approaches the asymptotes from the left and decreases without bound as it approaches from the right. The graph highlights the periodicity, symmetry, and vertical asymptotes of the tangent function

Transformations of Circular Functions

Sin and Cos

Transformations of a sine function can be represented by the general equation:

\[y=a\sin(nx-b)+c\]

Due to the similarity between sine and cosine functions, transformations of \(y = \cos\left(x\right)\) to \(y=a\cos(nx-b)+c\) have the same effects as transformations of \(y = \sin(x)\) to \(y = a\sin(nx - b) +c\). This is because the graph of \(y = \cos\left(x\right)\) is simply the graph of \(y = \sin\left(x\right)\) shifted \(\frac{\pi}{2}\) units in the negative direction of the \(x\)-axis. Thus, we will review only transformations of sine functions.

Reflection

Values of \(a<0\) in the general equation \(y=a\sin(nx-b)+c\) will cause reflections in the \(x\)-axis. The peaks and troughs of \(y=-\sin(x)\) will be opposite to those of \(y=\sin(x)\).

A graph comparing the sine function, y = sin(x) (blue), and its reflection, y = -sin(x) (green). Both functions have an amplitude of 1 and a period of 2π, oscillating between -1 and 1. The green curve is a vertical reflection of the blue curve across the x-axis, demonstrating the effect of a negative coefficient on the sine function. A magenta dashed arrow indicates the reflection and labels it as occurring when a < 0. The graph highlights the symmetry and transformation properties of the sine functionTypically, we also describe reflections about the \(y\)-axis. This can be represented by \(n<0\), such that \(y=\sin(-x)\) is a reflection across the \(y\)-axis of \(y=\sin(x)\). However, the symmetry of circular functions means that this reflection is identical to the reflection about the \(x\)-axis. That is,

\[\sin(-x)=-\sin(x)\]

Dilation

Dilation from the \(x\)-axis, that is, a vertical stretching or shrinking, is caused by changing the value of \(a\) in in the general equation \(y=a\sin(nx-b)+c\). When \(a > 1\) the amplitude of the function will get larger. For \(0 < a < 1\) the amplitude of the function will get smaller. If \(a < 0\) this results in a reflection of the graph about the \(x\)-axis, as described above.

This transformation of \(y = \sin(x)\) to \(y = a\sin(x)\) changes the height of the wave or amplitude. The amplitude is the distance from the mean position to the highest point on the sine wave.

Dilation from the \(y\)-axis, that is, a horizontal stretching or shrinking, is caused by changing the value of \(n\) in the general equation \(y=a\sin(nx-b)+c\). A value of \(n > 1\) results in a compression of the graph. For \(0 <n <1\) the graph stretches.

This transformation of \(y = \sin(x)\) to \(y = \sin(nx)\) changes the period of the function, that is, the wavelength. One full cycle of \(\sin(x)\) or \(\cos(x)\) occurs every \(2\pi\). Dilation from the \(y\)-axis varies this period. The period is given by \(\frac{2\pi}{n}\). Therefore, a value of \(n > 1\) decreases the period, while for \(0 <n <1\) the period increases.

Translation

Vertical translation is given by \(c\) in the general equation \(y=a\sin(nx-b)+c\). For values of \(c > 0\) the function is translated in the positive \(y\) direction. For \(c < 0\) the function is translated in the negative \(y\) direction.

This transformation of \(y = \sin(x)\) to \(y = \sin(x) + c\) changes the mean value of the function. For \(y  =\sin(x), c = 0\) and so the graph oscillates above and below the \(x\)-axis. For \(y=\sin(x)+c\) the graph oscillates around \(y=c\).

Horizontal translation is given by \(b\) in the general equation \(y=a\sin(nx-b)+c\). For values of \(b > 0\) the function is translated in the positive \(x\) direction. For \(b < 0\) the function is translated in the negative \(x\) direction.

This transformation of \(y = \sin(x)\) to  \(y = \sin(x - b)\) causes a phase shift of the graph, shifting the curve along the horizontal.

Tan

Transformations of tangent functions are similar to those of \(\sin\) and \(\cos\) functions. They can be represented by the general equation

\[y=a\tan(nx-b)+c\]

Review these transformations by clicking on the images below.

Summary of Transformations of Circular Functions

Sine, cosine and tangent functions are transformed according to the general equations:

  • \(y = a\sin(nx - b) + c\)
  • \(y = a\cos(nx - b) + c\)
  • \(y = a\tan(nx - b) + c\)
VariableTransformation Effect on graph
\(a\) Dilation of factor \(a\) from the \(x\)-axis

If \(a > 1\) the function is vertically stretched.
If \(0 < a < 1\) the function is vertically compressed.

For negative values, this indicates a reflection in the \(x\)-axis.

\(n\) Dilation of factor \(\frac{1}{n}\) from the \(y\)-axis If \(n > 1\) the function will be horizontally compressed (have more cycles/waves within \(2\pi\).
If \(0 < n < 1\) the function will be horizontally stretched (have less cycles/waves within \(2\pi\).
If  \(n<0\) the graph will be reflected about the \(y\)-axis.     
\(b\) Translation parallel to \(x\)-axis If \(b > 0\) the function will be translated in the positive \(x\) direction, while \(b < 0\) will cause translation in the positive \(x\) direction.
\(c\) Translation parallel to \(y\)-axis If \(c > 0\) the function will be translated in the positive \(y\) direction, while \(c < 0\) will cause translation in the negative \(y\) direction.

Determining Rules

Understanding the transformations can be used to determine the rule of an unknown circular function.

Worked Example

A function \(f(x)\) in the form \(f(x) = a\cos(nx), x\in[0,2\pi]\) is plotted as

f(x)=axos(nx), where x is between [0, 2pi]. The max value=2, min value=-2. The first two x-intercepts are pi/6 and pi/2.

Use the graph to determine the values \(a\) and \(n\), and the equation of the function.

Amplitude: max - mean position = \(2 - 0\); hence \(a = 2\)
Period: One full wavelength. One full wavelength is \(\frac{2\pi}{3}\)

As Period \(= \frac{2\pi}{n} = \frac{2\pi}{3}\), then \(n = 3\)

The equation of the unknown graph is \(f(x) = 2\cos(3x),  x\in[0,2]\).


Worked Example

A function with rule \(y = a\sin(nx) + c\) has the following properties:

Range \([1,9]\) and period \(\frac{1}{2}\).

Find the values of \(a, n\) and \(c\) and sketch the graph.

\(c\) is the mean position which is the middle of the range hence \(c = 5\).
\(a\) is the amplitude which is the max-mean, 9-5 hence \(a = 4\)

As Period \(= \frac{2\pi}{n} = \frac{1}{2}\), then \(n = 4\pi\)

\(\therefore y= 4\sin\left(4\pi{x}\right) + 5\)

Sketching the graph would give

Graph of y=4sin(4pix)+5


Modelling and Application

Circular functions describe relationships between variables that are cyclical or periodic in nature. Applications of circular functions extends into a wide range of real life situations which are sinusoidal in shape, that is, that can be described using \(y = a\sin(nx -b) + c\) or \(y = a\cos(nx - b) + c\).

Worked Example

The temperature, \(T^{\circ}C\), inside a building on a given day can be modelled by the equation

\(T = 20 + 7\sin\left(\frac{\pi{t}}{5}\right)\), where \(t\) is the number of hours after 9 am.

  1. What is the temperature at 9 am?

    2. Let \(t = 0,\quad T = 20 + 7\sin\left(0\right)\)

    \(T = 20^{\circ}C\)

    The temperature at 9am is \(20^{\circ}C\).

  2. What is the maximum temperature in the building?

    3. Max temp is amplitude + mean position

    \[\begin{align}T &= 20 + 7 \\ T &= 27^{\circ}C\end{align}\]

    The maximum temperature is \(27^{\circ}C\).

  3. At what time does the maximum temperature first occur?

    4. Max temp occurs when \(\sin(x)\) is at maximum. This occurs at \(\sin(\frac{\pi}{2})\)

    Solve \(\sin\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi{t}}{5}\right)\) for \(t\)

    \(\begin{align} \dfrac{\pi}{2} &= \dfrac{\pi{t}}{5} \\ t &= \frac{5}{2} \text{ or } 2.5\text{ hours} \end{align}\)

    As \(t\) is measured in hours after 9am, the maximum temperature occurs at 11:30am.

  4. Sketch the graph of \(T\) against \(t\) for \(t\in[0,20]\)

    5. \(a = 7; c = 20\)

    Range: \([13,27]\); Period \(= \frac{2\pi}{n} = \frac{2\pi}{\pi/5} = 10\)

    Graph of T = 20+7sin(pit/5) between t = 0 and t =20. Shows a sine wave with y-intercept (0, 20)