9.01 Key features of sine, cosine, and tangent graphs
Graphing $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ
The definitions of the trigonometric functions $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ that we've seen use the ratios of side lengths of a right triangle.
For a right triangle, where $\theta$θ is the measure for one of the angles (excluding the right angle), we have that:
| $\sin\theta$sinθ | $=$= | $\frac{\text{opposite }}{\text{hypotenuse }}$opposite hypotenuse |
| $\cos\theta$cosθ | $=$= | $\frac{\text{adjacent }}{\text{hypotenuse }}$adjacent hypotenuse |
| $\tan\theta$tanθ | $=$= | $\frac{\text{opposite }}{\text{adjacent }}$opposite adjacent |
Now recall the unit circle where any point on the circle has $x=\cos\theta$x=cosθ and $y=\sin\theta$y=sinθ, where $\theta$θ is the angle with the positive $x$x-axis.
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| A right triangle inscribed in the unit circle. |
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| Definition of $\cos$cos and $\sin$sin can extend beyond $0^\circ\le\theta\le90^\circ$0°≤θ≤90°. |
As we move through different values of $\theta$θ the value of $\cos\theta$cosθ and $\sin\theta$sinθ move accordingly between $-1$−1 and $1$1. If we plot the values of $\cos\theta$cosθ and $\sin\theta$sinθ according to different values of $\theta$θ on the unit circle, we get the following graphs:
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| $y=\cos\theta$y=cosθ |
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| $y=\sin\theta$y=sinθ |
Worked example
Question 1
By using the graph of $y=\cos x$y=cosx, what is the sign of $\cos345^\circ$cos345°?
Think: Using the graph of $y=\cos x$y=cosx, we can roughly estimate where the point $\left(345^\circ,\cos345^\circ\right)$(345°,cos345°) lies and from this, determine the sign of $\cos345^\circ$cos345°.
Do: We plot the point on the graph of $y=\cos x$y=cosx below.
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The point $\left(345^\circ,\cos345^\circ\right)$(345°,cos345°) drawn on the graph of $y=\cos x$y=cosx. |
We can observe that the height of the curve at this point is above the $x$x-axis, and that $\cos345^\circ$cos345° is positive.
Practice questions
question 2
Consider the equation $y=\sin x$y=sinx.
Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin150^\circ$sin150°?
Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin210^\circ$sin210°?
Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin330^\circ$sin330°?
Complete the table of values giving answers in exact form.
$x$x $0^\circ$0° $30^\circ$30° $90^\circ$90° $150^\circ$150° $180^\circ$180° $210^\circ$210° $270^\circ$270° $330^\circ$330° $360^\circ$360° $\sin x$sinx $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the graph of $y=\sin x$y=sinx.
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question 3
Consider the equation $y=\cos x$y=cosx.
Complete the table of values, giving answers in exact form.
$x$x $0^\circ$0° $60^\circ$60° $90^\circ$90° $120^\circ$120° $180^\circ$180° $240^\circ$240° $270^\circ$270° $300^\circ$300° $360^\circ$360° $\cos x$cosx $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the graph of $y=\cos x$y=cosx.
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Key features of sine and cosine curves
The graphs of $y=\cos\theta$y=cosθ and $y=\sin\theta$y=sinθ have many properties. Each graph demonstrates repetition. We call the graphs of $y=\cos\theta$y=cosθ and $y=\sin\theta$y=sinθ periodic and define a cycle as any section of the graph that can be translated to complete the rest of the graph. We also define the period as the length of one cycle. For both graphs, the period is $360^\circ$360°.
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| An example of a cycle |
Because of the oscillating behavior, both graphs have regions where the curve is increasing and decreasing. Remember that we say the graph of a particular curve is increasing if the $y$y-values increase as the $x$x-values increase. Similarly, we say the graph is decreasing if the $y$y-values decrease as the $x$x-values increase.
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| An example of where $y=\sin x$y=sinx is decreasing |
In addition, the height of each graph stays between $y=-1$y=−1 and $y=1$y=1 for all values of $\theta$θ, since each coordinate of a point on the unit circle can be at most $1$1 unit from the origin. We call half the distance between the maximum and minimum the amplitude. For the basic graphs, the amplitude is $1$1.
Key Features of $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ in degrees
| $y=\sin\theta$y=sinθ | $y=\cos\theta$y=cosθ | |
|---|---|---|
| Domain: | $x$x is any real number | $x$x is any real number |
| Range: | $y\in[-1,1]$y∈[−1,1] | $y\in[-1,1]$y∈[−1,1] |
| $x$x-intercepts: | $x=0^\circ+180^\circ n$x=0°+180°n, $n$n an integer | $x=90^\circ+180^\circ n$x=90°+180°n, $n$n an integer |
| $y$y-intercept: | $y=0$y=0 | $y=1$y=1 |
| Period: | $360^\circ$360° | $360^\circ$360° |
| Amplitude: | $1$1 | $1$1 |
Practice questions
QUESTION 4
Consider the curve $y=\sin x$y=sinx drawn below and answer the following questions.
What is the $y$y-intercept? Give your answer as coordinates in the form $\left(a,b\right)$(a,b).
What is the maximum $y$y-value?
What is the minimum $y$y-value?
question 5
Consider the curve $y=\cos x$y=cosx drawn below and determine whether the following statements are true or false.
The graph of $y=\cos x$y=cosx is cyclic.
True
AFalse
BAs $x$x approaches infinity, the height of the graph approaches infinity.
True
AFalse
BThe graph of $y=\cos x$y=cosx is increasing between $x=90^\circ$x=90° and $x=180^\circ$x=180°.
True
AFalse
B
question 6
Consider the curve $y=\cos x$y=cosx drawn below and answer the following questions.
If one cycle of the graph of $y=\cos x$y=cosx starts at $x=\left(-90\right)^\circ$x=(−90)°, when does the next cycle start?
In which of the following regions is the graph of $y=\cos x$y=cosx decreasing? Select all that apply.
$\left(-180\right)^\circ
A$\left(-360\right)^\circ
B$0^\circ
C$180^\circ
DWhat are the $x$x-values of the $x$x-intercepts in the region $0^\circ
Graphing $y=\tan\theta$y=tanθ
The behavior of tangent curves (often abbreviated to tan) is quite different to sine and cosine.
Remember:
- $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ
- The $\sin$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 $\tan\theta$tanθ is a function obtained through the division of $\sin\theta$sinθ by $\cos\theta$cosθ, 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 periodic function, $y=\cos\theta$y=cosθ has the value $0$0 quite often, wherever $\theta=\frac{\pi}{2}+\pi n$θ=π2+πn, $n$n an integer.
$y=\cos\theta$y=cosθ
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.
Exploration
What does $y=\tan\theta$y=tanθ look like between asymptotes?
The following applet will help us discover the graph of $y=\tan x$y=tanx
Check the box for "Show $y=\tan x$y=tanx" and then drag the point at the origin to the left and right. Watch how the corresponding points of sine and cosine change and how this creates the graph of the tangent curve.
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 $\sin\theta=\cos\theta$sinθ=cosθ and thus at that point the value of $\tan\theta$tanθ 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 getting larger and larger.
So now we have created the graph of $y=\tan x$y=tanx
Practice question
Question 7
Consider the equation $y=\tan x$y=tanx.
Using the fact that $\tan45^\circ=1$tan45°=1, what is the value of $\tan135^\circ$tan135°?
Using the fact that $\tan45^\circ=1$tan45°=1, what is the value of $\tan225^\circ$tan225°?
Using the fact that $\tan45^\circ=1$tan45°=1, what is the value of $\tan315^\circ$tan315°?
Complete the table of values, giving answers in exact form.
$x$x $0$0 $45^\circ$45° $90^\circ$90° $135^\circ$135° $180^\circ$180° $\tan x$tanx $0$0 $\editable{}$ undefined $\editable{}$ $\editable{}$ Complete the table of values, giving answers in exact form.
$x$x $180^\circ$180° $225^\circ$225° $270^\circ$270° $315^\circ$315° $360^\circ$360° $\tan x$tanx $0$0 $\editable{}$ undefined $\editable{}$ $\editable{}$ Plot the graph of $y=\tan x$y=tanx.
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Key features of tangent curves
In general, 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.
Tangent curves in degrees
Key features of $y=\tan\theta$y=tanθ, in both degrees and radians:
| Degrees | Radians | |
|---|---|---|
| Domain: | $x\ne90^\circ+180^\circ n$x≠90°+180°n | $x\ne\frac{\pi}{2}+\pi n$x≠π2+πn |
| Range: | $y$y any real number | $y$y any real number |
| $x$x-intercepts: | $x=0^\circ+180^\circ n$x=0°+180°n, $n$n an integer | $x=0+\pi n$x=0+πn, $n$n an integer |
| $y$y-intercept: | $y=0$y=0 | $y=0$y=0 |
| Period: | $180^\circ$180° | $\pi$π |
Practice Questions
question 8
Consider the graph of $y=\tan x$y=tanx for $-2\pi\le x\le2\pi$−2π≤x≤2π.
How would you describe the graph?
Periodic
ADecreasing
BEven
CLinear
DWhich of the following is not appropriate to refer to in regard to the graph of $y=\tan x$y=tanx?
Amplitude
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=tanx in radians.
State the range of $y=\tan x$y=tanx.
$-\infty
A$y>0$y>0
B$\frac{-\pi}{2}
C$-\pi
DAs $x$x increases, what would be the next asymptote of the graph after $x=\frac{7\pi}{2}$x=7π2?
question 9
Given the unit circle, which two of the following is true about the graph of $y=\tan x$y=tanx?
The graph of $y=\tan x$y=tanx repeats in regular intervals since the values of $\sin x$sinx and $\cos x$cosx repeat in regular intervals.
AThe graph of $y=\tan x$y=tanx is defined for any measure of $x$x.
BSince the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$−1≤y≤1.
CThe range of values of $y=\tan x$y=tanx is $-\infty
D