NZ Level 8 (NZC) Level 3 (NCEA) [In development] Key features of tangent curves
Lesson

The behaviour of tangent curves (often abbreviated to tan) are not like any of the functions we have previously studied.

Some basics -

• remember that $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ
• remember also that $\sin$sin function related to a mapping of the $y$y-coordinates as we moved around the unit circle, and similarly the $\cos$cos function related to a mapping of the $x$x-coordinates.

As $\tan$tan is a function obtained through the division of $\sin$sin by $\cos$cos, we will have a problem with the tangent values wherever $\cos=0$cos=0 because a fraction with $0$0 on the denominator is undefined. Unfortunately, because it's a cyclic function, $y=\cos x$y=cosx has the value $0$0 quite often, as shown below... This cosine curve has $0$0 value at $\frac{\pi}{2}$π2, $\frac{3\pi}{2}$3π2 and multiplies of $2\pi$2π of these.

This means that at all of these values the function $y=\tan x$y=tanx will not exist.  In fact this function will have vertical asymptotes at those points.

But what about the value of tan between these asymptotes?

The following applet will help us discover the graph of $y=\tan x$y=tanx

Drag the point at the origin to the left and right. Watch how the corresponding points of sine and cosine change.

They start off with sine very small and cosine $1$1, so sine $\div$÷​ cosine is a very small value as well.  As the point moves to the right ($x$x values get larger) the sine values increase and the cosine values decrease, up to the point where $\sin=\cos$sin=cos and thus at that point the value of $\tan$tan would be $1$1.  This happens at $\frac{\pi}{4}$π4.  Continue along now and we see that the value of sine continues to grow and the values of cosine get very small.  We know that when we have a fraction, if the denominator gets smaller and smaller and smaller then the value of the fraction is gettting larger and larger.

Now, turn on the TRACE, ("show" tan button).  Repeat these steps and you will see the tangent curve appear.  Talk it through yourself as this will help you to understand this important connection between sine, cosine and tangent functions.

So now we have created the graph of $y=\tan x$y=tanx ## Key features of tangent curves

Tangent curves have

• vertical asymptotes
• a period, it is the distance between the vertical asymptotes
• same general shape (although transformations can be applied)
• multiple solutions (places where the curve crossed the $x$x-axis)
• $1$1 $y$y-intercept, as long as the asymptote is not the axis.

#### Worked Examples

##### question 1

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

1. How would you describe the graph?

Periodic

A

Decreasing

B

Even

C

Linear

D

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

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

$-\piπ<y<π D$-\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?

##### question 2

Consider the expression $\tan\theta$tanθ.

1. $\tan\theta$tanθ is defined as $\frac{\text{opposite }}{\text{adjacent }}$opposite adjacent for $0\le\theta<\frac{\pi}{2}$0θ<π2 in a right-angled triangle. What happens to the value of $\tan\theta$tanθ as $\theta$θ increases from $0$0 to $\frac{\pi}{2}$π2?

The value of $\tan\theta$tanθ decreases towards $0$0.

A

The value of $\tan\theta$tanθ becomes larger and larger.

B

The value of $\tan\theta$tanθ decreases towards $0$0.

A

The value of $\tan\theta$tanθ becomes larger and larger.

B
2. Express $\tan\theta$tanθ in terms of $\sin\theta$sinθ and $\cos\theta$cosθ.

3. The graph of $y=\cos x$y=cosx for $-2\pi\le x\le2\pi$2πx2π is provided. For what values of $x$x is $\cos x=0$cosx=0?

List the values on the same line, separated by a comma.

4. Hence, for what values of $x$x between $-2\pi$2π and $2\pi$2π is $\tan x$tanx undefined?

List the values on the same line, separated by a comma.

5. By relating angles in the second, third and fourth quadrant to angles in the first quadrant, complete the table below.

 $x$x $\tan x$tanx $-2\pi$−2π $-\frac{7\pi}{4}$−7π4​ $-\frac{5\pi}{4}$−5π4​ $-\pi$−π $-\frac{3\pi}{4}$−3π4​ $-\frac{\pi}{4}$−π4​ $0$0 $\frac{\pi}{4}$π4​ $\frac{3\pi}{4}$3π4​ $\pi$π $\frac{5\pi}{4}$5π4​ $\frac{7\pi}{4}$7π4​ $2\pi$2π $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
6. Graph $y=\tan\theta$y=tanθ for $-2\pi\le\theta\le2\pi$2πθ2π.

##### question 3

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

3. Determine the range of the function.

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

A

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

B

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

C

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

D

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

A

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

B

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

C

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

D

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions