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7.08 Circular functions: tangent

Lesson

Recall from the introduction of the unit circle we can define the tangent function as follows:

  • The tangent of the angle can be geometrically defined to be $y$y-coordinate of point $Q$Q, where $Q$Q is the intersection of the extension of the line $OP$OP and the tangent of the circle at $\left(1,0\right)$(1,0)
  • Using similar triangles we can also define this algebraically as the ratio: $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ
  • This also represents the gradient of the line that forms the angle $\theta$θ to the positive $x$x-axis

Let's revisit the applet from the introduction once more and focus our attention on the orange line. With the definitions above in mind, for what values of $\theta$θ is $\tan\theta$tanθ undefined? This would be the same as asking when is the slope of the line $OP$OP undefined, or when does $\cos\theta=0$cosθ=0. What is the value of $\tan0^\circ$tan0°? How often does the pattern repeat? That is what is the period of $\tan\theta$tanθ?

Change the angle in the applet below and try to imagine the shape of the graph of $\tan\theta$tanθ and key features.

 

The graph of $\tan\theta$tanθ


 

Key features

Key features of the graph of $y=\tan\left(\theta\right)$y=tan(θ) are:

  • Asymptotes: vertical asymptotes appear at $\theta=\frac{\pi}{2}+n\pi$θ=π2+nπ, for $n$n any integer
  • Axis intercepts: vertical axis intercept: $\left(0,0\right)$(0,0), horizontal axis intercepts at: $\theta=n\pi$θ=nπ, for $n$n any integer - at each intercept there is a point of inflection
  • Period: The period of this graph can be found as the distance between two successive asymptotes. Hence, the period is: $\pi$π
  • Range: This graph is unbounded. Hence, the range is: $\left(-\infty,\infty\right)$(,)
  • Domain: The function is undefined at each vertical asymptote. Hence, the domain is:  $\theta:\theta\in\Re,\theta\ne n\pi$θ:θ,θnπ, where $n$n is any integer
  • Key points: useful for graphing, the function passes through $\left(\frac{\pi}{4},1\right)$(π4,1) and $\left(\frac{-\pi}{4},-1\right)$(π4,1)
  • Symmetry: The tangent graph is an odd function, that is $\tan\left(-\theta\right)=-\tan\theta$tan(θ)=tanθ. This also means the graph has point symmetry about the origin (or any point of inflection) by $180^\circ$180°

The asymptotes of the function are where the angle $\theta$θ would cause the line $OP$OP to be vertical and hence the gradient is undefined. We can also see this through the definition $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ. The function is undefined where $\cos\theta=0$cosθ=0 and the graph approaches the vertical lines at these values forming asymptotes.

The fact that the tangent function repeats at intervals of $\pi$π can be verified by considering the unit circle diagram. Either by considering the gradient of the line $OP$OP and how the gradient will be the same for a point at an angle of $\theta$θ or at at angle of $\theta+\pi$θ+π. Or we can consider this algebraically by looking at the ratio $\frac{\sin\theta}{\cos\theta}$sinθcosθ. If $\pi$π is added to an angle $\theta$θ, then the diagram  below shows that $\sin(\theta+\pi)$sin(θ+π) has the same magnitude as $\sin\theta$sinθ but opposite sign. The same relation holds between $\cos(\theta+\pi)$cos(θ+π) and $\cos\theta$cosθ.

We make use of the definition: $$

$\tan(\theta+\pi)$tan(θ+π) $=$= $\frac{\sin(\theta+\pi)}{\cos(\theta+\pi)}$sin(θ+π)cos(θ+π)
  $=$= $\frac{-\sin\theta}{-\cos\theta}$sinθcosθ
  $=$= $\tan\theta$tanθ

 

Practice question

question 1

Consider the graph of $y=\tan x$y=tanx for $-2\pi\le x\le2\pi$2πx2π.

Loading Graph...

  1. How would you describe the graph?

    Periodic

    A

    Decreasing

    B

    Even

    C

    Linear

    D
  2. Which of the following is not appropriate to refer to in regard to the graph of $y=\tan x$y=tanx?

    Amplitude

    A

    Range

    B

    Period

    C

    Asymptotes

    D
  3. The period of a periodic function is the length of $x$x-values that it takes to complete one full cycle.

    Determine the period of $y=\tan x$y=tanx in radians.

  4. State the range of $y=\tan x$y=tanx.

    $-\infty<y<

    A

    $y>0$y>0

    B

    $\frac{-\pi}{2}π2<y<π2

    C

    $-\piπ<y<π

    D
  5. As $x$x increases, what would be the next asymptote of the graph after $x=\frac{7\pi}{2}$x=7π2?

 

Transformations of $\tan x$tanx

Just as we transformed the trigonometric functions $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ we can apply parameters to the equation $y=\tan x$y=tanx to transform it to $y=a\tan\left(b\left(x-c\right)\right)+d$y=atan(b(xc))+d.

Use the geogebra applet below to adjust the parameters in $y=a\tan\left(b\left(x-c\right)\right)+d$y=atan(b(xc))+d and observe how it affects the graph. Try to answer the following questions.

  • Which parameters affect the position of the vertical asymptotes? Which ones don't?
  • Which parameters translate the graph, leaving the shape unchanged? Which ones affect the size?
  • Which parameter changes the period of the graph? Does making this parameter larger make the period larger?

The constants $a$a, $b$b, $c$c and $d$d transform the tangent graph. Let's summarise the impact of each:

Summary

To obtain the graph of $y=a\tan\left(b\left(x-c\right)\right)+d$y=atan(b(xc))+d from the graph of $y=\tan\left(x\right)$y=tan(x):

  • $a$a dilates(stretches) the graph by a factor of $a$a from the $x$x-axis. Before applying translations, this will cause the point $\left(\frac{\pi}{4},1\right)$(π4,1) to stretch to $\left(\frac{\pi}{4},a\right)$(π4,a) and similarly the point $\left(\frac{-\pi}{4},-1\right)$(π4,1)will stretch to $\left(\frac{-\pi}{4},-a\right)$(π4,a) 
  • When $a<0$a<0 the graph was reflected about the $x$x-axis. So the graph will be decreasing between the asymptotes rather than increasing
  • $b$b dilates(stretches) the graph by a factor of $\frac{1}{b}$1b from the $y$y-axis. Hence, the period becomes: $period=\frac{\pi}{b}$period=πb
  • When $b<0$b<0 the graph was reflected about the $y$y-axis
  • $c$c translates the graph $c$c units horizontally
  • $d$d translates the graph $d$d units vertically

 

 

Worked examples

Example 1

Illustrate change in dilation of the graph $y=\tan x$y=tanx by sketching the graphs of $y=\tan x$y=tanx together with $y=3\tan x$y=3tanx and $y=\frac{2}{7}\tan x$y=27tanx.

Note since only a vertical dilation has been applied all three graphs will share $x$x-intercepts and vertical asymptotes. We first sketch the base graph of $y=\tan x$y=tanx, shown in blue in the graph below. This graph will go through the points $\left(\frac{\pi}{4},1\right)$(π4,1)$\left(0,0\right)$(0,0) and $\left(\frac{-\pi}{4},-1\right)$(π4,1). The asymptotes will be located at $\frac{\pi}{2}$π2, $\frac{-\pi}{2}$π2, $\frac{3\pi}{2}$3π2, $\frac{-3\pi}{2}$3π2,....

The graph of $y=3\tan x$y=3tanx has a vertical dilation by a factor of $3$3. So we can plot the points $\left(\frac{\pi}{4},3\right)$(π4,3) and $\left(\frac{-\pi}{4},-3\right)$(π4,3) to show this stretch clearly. Similarly the graph $y=\frac{2}{7}\tan x$y=27tanx has a vertical dilation by a factor of $\frac{2}{7}$27. To show this we can plot the points $\left(\frac{\pi}{4},\frac{2}{7}\right)$(π4,27) and $\left(\frac{-\pi}{4},\frac{-2}{7}\right)$(π4,27). The graphs together are shown below.

 

 

 

Example 2

Sketch the graph of $f(x)=\tan\left(x-\frac{\pi}{4}\right)$f(x)=tan(xπ4).

We see that the function $\tan x$tanx has been moved to the right a distance $\frac{\pi}{4}$π4. We can sketch this by graphing the base graph of $y=\tan\theta$y=tanθ and shifting each point right by $\frac{\pi}{4}$π4.

The phase shift will also move the asymptotes and since $\tan x$tanx is undefined at $x=\frac{\pi}{2}+n\pi$x=π2+nπ for all integers $n$n, the undefined points for $\tan\left(x-\frac{\pi}{4}\right)$tan(xπ4) must be $x=\frac{3\pi}{4}+n\pi$x=3π4+nπ.  The graph is shown below in purple.

 

Practice questions

question 2

Consider the function $y=-4\tan\frac{1}{5}\left(x+\frac{\pi}{4}\right)$y=4tan15(x+π4).

  1. Determine the period of the function, giving your answer in radians.

  2. Determine the phase shift of the function, giving your answer in radians.

  3. Determine the range of the function.

    $[-1,1]$[1,1]

    A

    $(-\infty,0]$(,0]

    B

    $[0,\infty)$[0,)

    C

    $(-\infty,\infty)$(,)

    D

question 3

The graph of $y=\tan x$y=tanx is shown below. On the same set of axes, draw the graph of $y=5\tan x$y=5tanx.

  1. Loading Graph...

question 4

Select all functions that have the same graph as $y=-\tan x$y=tanx.

  1. $y=-\tan\left(x+\frac{3\pi}{4}\right)$y=tan(x+3π4)

    A

    $y=-\tan\left(x+\pi\right)$y=tan(x+π)

    B

    $y=-\tan\left(x+\frac{\pi}{2}\right)$y=tan(x+π2)

    C

    $y=-\tan\left(x+2\pi\right)$y=tan(x+2π)

    D

Sketching tips

We can graph a tangent function by graphing the base function and then applying the transformations in stages to achieve the final sketch. Just as with sine and cosine functions we could also sketch the function using key features. The following steps may assist in sketching:

  1. Ensure you formula is in the required format, $y=a\tan\left(b\left(x-c\right)\right)+d$y=atan(b(xc))+d rearrange if necessary. Write down $a$a, $b$b, $c$c and $d$d.
  2. Sketch in a dotted line for $y=d$y=d. This is the central line where the points of inflection will lie. If the graph has not been vertically translated this will be the $x$x-axis.
  3. Plot the point $\left(c,d\right)$(c,d). This is where the point of inflection at the origin of $y=\tan x$y=tanx has been translated to.
  4.  Find the period: $P=\frac{\pi}{b}$P=πb .
  5. From the point $\left(c,d\right)$(c,d) draw dotted lines for the asymptotes half a period in both directions. That is at $x=c+\frac{P}{2}$x=c+P2 and  $x=c-\frac{P}{2}$x=cP2. Then at integer multiples of the period, $P$P, in each direction from these asymptotes until you have the required domain.
  6. To achieve a more accurate sketch and clearly show the dilation, plot the points $\left(c+\frac{P}{4},d+a\right)$(c+P4,d+a)  and $\left(c-\frac{P}{4},d-a\right)$(cP4,da) 
  7. Draw a smooth curve through your points and approaching the asymptotes.
  8. Repeat this smooth curve for each period.

 

Worked example

Example 3

Sketch the function $y=3\tan\left(2\left(x-\frac{\pi}{4}\right)\right)+1$y=3tan(2(xπ4))+1 for the interval $-\pi\le x\le\pi$πxπ

Think: What transformations would take $y=\tan x$y=tanx to $y=3\tan\left(2\left(x-\frac{\pi}{4}\right)\right)+1$y=3tan(2(xπ4))+1

  • We would dilate the graph by a factor of 3 from the $x$x-axis
  • We need to dilate the graph by a factor of $\frac{1}{2}$12 horizontally. Hence, the period becomes $\frac{\pi}{2}$π2.
  • We need to translate the graph by $1$1 unit vertically
  • We need to translate the graph by $\frac{\pi}{4}$π4 units horizontally

Do: List the parameters $a=3$a=3$b=2$b=2$c=\frac{\pi}{4}$c=π4 and $d=1$d=1. Sketch a dotted line for the central line $y=1$y=1 and plot the point $\left(c,d\right)=\left(\frac{\pi}{4},1\right)$(c,d)=(π4,1)

Find the period: $period=\frac{\pi}{b}$period=πb$=\frac{\pi}{2}$=π2 and draw dotted lines for the asymptotes half a period in both directions from the point $\left(\frac{\pi}{4},1\right)$(π4,1). Then repeat at multiples of the period from these lines.

From the point $\left(\frac{\pi}{4},1\right)$(π4,1) plot a point by going forwards $\frac{period}{4}$period4$=\frac{\pi}{8}$=π8 and up $a$a units ($3$3 units). Mirror this by plotting a second point backwards $\frac{\pi}{8}$π8 from $\left(\frac{\pi}{4},1\right)$(π4,1) and down $3$3 units.

Join the points with a smooth curve which also approaches the asymptotes. 

Lastly, repeat the pattern for each period.

Graph of $y=3\tan\left(2\left(x-\frac{\pi}{4}\right)\right)+1$y=3tan(2(xπ4))+1

Reflect: Does the graph match how it should look after transformations? Does its cycle repeat the correct number of times for the domain given?

Axis intercepts

For our example above we did not have a $y$y-intercept as an asymptote coincided with the $y$y-axis. If this was not the case we could find the $y$y-intercept by evaluating the function at $x=0$x=0.

If we were required to label the $x$x-intercepts and there had been no vertical translation these would be found at the points of inflection at $x=c+Pn$x=c+Pn, where $P$P in the period and $n$n is any integer. However, if as in our example we have a vertical shift the $x$x-intercepts can be found with the assistance of technology or by solving the equation when $y=0$y=0. We will look further at solution to trigonometric equations later in this chapter. 

Practice questions

Question 5

Consider the function $y=-\tan x$y=tanx.

Use radians to answer the following questions, where appropriate.

  1. Determine the $y$y-intercept.

  2. How far apart are the asymptotes of the function?

  3. Hence determine the period of the function.

  4. State the first asymptote of the function for $x\ge0$x0.

  5. State the first asymptote of the function for $x\le0$x0.

  6. By moving the three given points, graph the function.

    Loading Graph...

Question 6

Consider the function $y=\tan\left(x-\frac{\pi}{2}\right)$y=tan(xπ2).

Answer the following questions in radians, where appropriate.

  1. Determine the $y$y-intercept.

    $\left(0,\frac{\pi}{2}\right)$(0,π2)

    A

    $\left(0,0\right)$(0,0)

    B

    $\left(0,-\pi\right)$(0,π)

    C

    There is no $y$y-intercept.

    D
  2. Determine the period of the function.

  3. How far apart are the asymptotes of the function?

  4. State the first asymptote of the function for $x>0$x>0.

  5. State the first asymptote of the function for $x\le0$x0.

  6. By moving the three given points, graph the function.

    Loading Graph...

Question 7

Consider the function $y=\tan4x-3$y=tan4x3.

Answer the following questions in radians, where appropriate.

  1. Determine the $y$y-intercept.

  2. Determine the period of the function.

  3. How far apart are the asymptotes of the function?

  4. State the first asymptote of the function for $x\ge0$x0.

  5. State the first asymptote of the function for $x\le0$x0.

  6. Graph the function.

    Loading Graph...

Outcomes

1.2.1.4

examine transformations of the graphs of 𝑓(𝑥), including dilations and reflections, and the graphs of 𝑦=𝑎f(𝑥) and 𝑦=𝑓(𝑏x), translations, and the graphs of 𝑦=𝑓(𝑥+𝑐) and 𝑦=𝑓(𝑥)+𝑑; 𝑎,𝑏,𝑐,𝑑 ∈ R

2.3.2.1

understand the unit circle definition of cos(𝜃), sin(𝜃)and tan(𝜃) and periodicity using radians

2.3.2.3

sketch the graphs of 𝑦=sin(𝑥), 𝑦=cos(𝑥) , and 𝑦=tan(𝑥) on extended domains

2.3.2.4

investigate the effect of the parameters 𝐴,𝐵,𝐶 and 𝐷 on the graphs of 𝑦=𝐴sin(𝐵(𝑥+𝐶))+𝐷, 𝑦=𝐴cos(𝐵(𝑥+𝐶))+𝐷 with and without technology

2.3.2.5

sketch the graphs of 𝑦=𝐴sin(𝐵(𝑥+𝐶))+𝐷, 𝑦=𝐴cos(𝐵(𝑥+𝐶))+𝐷 with and without technology

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