5.01 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. 

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.

A right triangle inscribed in the unit circle.

 

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:

$y=\cos\theta$y=cosθ

 

$y=\sin\theta$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.

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

question 3

 

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°.

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.

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.

 

Practice questions

QUESTION 4

question 5

question 6

 

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.

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

 

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.

 

Practice Questions

question 8

question 9