Trigonometric Graphs

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 $\tan\theta=\frac{\sin\theta}{\cos\theta}$
`t``a``n``θ`=`s``i``n``θ``c``o``s``θ` - remember also that $\sin$
`s``i``n`function related to a mapping of the $y$`y`-coordinates as we moved around the unit circle, and similarly the $\cos$`c``o``s`function related to a mapping of the $x$`x`-coordinates.

As $\tan$`t``a``n` is a function obtained through the division of $\sin$`s``i``n` by $\cos$`c``o``s`, we will have a problem with the tangent values wherever $\cos=0$`c``o``s`=0 because a fraction with $0$0 on the denominator is undefined. Unfortunately, because it's a cyclic function, $y=\cos x$`y`=`c``o``s``x` 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`=`t``a``n``x` 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`=`t``a``n``x`

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 $\sin=\cos$`s``i``n`=`c``o``s` and thus at that point the value of $\tan$`t``a``n` 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`=`t``a``n``x`

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.

Consider the graph of $y=\tan x$`y`=`t``a``n``x` for $-2\pi\le x\le2\pi$−2π≤`x`≤2π.

Loading Graph...

How would you describe the graph?

Periodic

ADecreasing

BEven

CLinear

DPeriodic

ADecreasing

BEven

CLinear

DWhich of the following is not appropriate to refer to in regard to the graph of $y=\tan x$

`y`=`t``a``n``x`?Amplitude

ARange

BPeriod

CAsymptotes

DAmplitude

ARange

BPeriod

CAsymptotes

DThe 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`=`t``a``n``x`in radians.State the range of $y=\tan x$

`y`=`t``a``n``x`.$-\infty

−∞< `y`<∞A$y>0$

`y`>0B$\frac{-\pi}{2}

−π2< `y`<π2C$-\pi

−π< `y`<πD$-\infty

−∞< `y`<∞A$y>0$

`y`>0B$\frac{-\pi}{2}

−π2< `y`<π2C$-\pi

−π< `y`<πDAs $x$

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

Consider the expression $\tan\theta$`t``a``n``θ`.

$\tan\theta$

`t``a``n``θ`is defined as $\frac{\text{opposite }}{\text{adjacent }}$opposite adjacent for $0\le\theta<\frac{\pi}{2}$0≤`θ`<π2 in a right-angled triangle.What happens to the value of $\tan\theta$

`t``a``n``θ`as $\theta$`θ`increases from $0$0 to $\frac{\pi}{2}$π2?The value of $\tan\theta$

`t``a``n``θ`decreases towards $0$0.AThe value of $\tan\theta$

`t``a``n``θ`becomes larger and larger.BThe value of $\tan\theta$

`t``a``n``θ`decreases towards $0$0.AThe value of $\tan\theta$

`t``a``n``θ`becomes larger and larger.BExpress $\tan\theta$

`t``a``n``θ`in terms of $\sin\theta$`s``i``n``θ`and $\cos\theta$`c``o``s``θ`.The graph of $y=\cos x$

`y`=`c``o``s``x`for $-2\pi\le x\le2\pi$−2π≤`x`≤2π is provided. For what values of $x$`x`is $\cos x=0$`c``o``s``x`=0?List the values on the same line, separated by a comma.

Loading Graph...Hence, for what values of $x$

`x`between $-2\pi$−2π and $2\pi$2π is $\tan x$`t``a``n``x`undefined?List the values on the same line, separated by a comma.

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

$x$ `x`$-2\pi$−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 $\tan x$ `t``a``n``x`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Graph $y=\tan\theta$

`y`=`t``a``n``θ`for $-2\pi\le\theta\le2\pi$−2π≤`θ`≤2π.Loading Graph...

Consider the function $y=-4\tan\frac{1}{5}\left(x+\frac{\pi}{4}\right)$`y`=−4`t``a``n`15(`x`+π4).

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

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

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

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