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iGCSE (2021 Edition)

7.03 Graphs of trigonometric functions

Lesson

The unit circle provides us with a visual understanding that the trigonometric functions of $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ exist for angles larger than what can be contained in a right-angled triangle.

The unit circle definitions of $\sin\theta$sinθ and $\cos\theta$cosθ tell us that the value of these functions will be the $x$x and $y$y-values respectively of a point on the unit circle after having rotated by an angle of measure $\theta$θ in the anticlockwise direction. Or, if $\theta$θ is negative, then the point is rotated in the clockwise direction.

Definition of $\cos\theta$cosθ and $\sin\theta$sinθ can extend beyond $0^\circ\le\theta\le90^\circ$0°θ90°.

 

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.

The animation below shows this process for $y=\sin\theta$y=sinθ as $\theta$θ travels around the unit circle.

Graphs of sin, cos and tan functions

If we plot the values of $\sin\theta$sinθ and $\cos\theta$cosθaccording to different values of $\theta$θ on the unit circle, we get the following graphs:

$y=\sin\theta$y=sinθ
$y=\cos\theta$y=cosθ

The simplest way to calculate $\tan\theta$tanθ is to use the values in the above graphs to evaluate $\frac{\sin\theta}{\cos\theta}$sinθcosθ, which gives us the following graph:

$y=\tan\theta$y=tanθ

 

Notice that all of these graphs are constructed with degrees on the horizontal axis. The function values behave in the same way as in the unit circle - for example, in the graph above of $y=\cos\theta$y=cosθ, we can see that it has negative $y$y-values for all of the angles in the domain of $90^\circ<\theta<180^\circ$90°<θ<180°. These are the values associated with the second quadrant, that is, the "S" in ASTC, where we know that $\cos\theta$cosθ will be negative. 

Properties of trigonometric graphs

The graphs of $y=\cos\theta$y=cosθ and $y=\sin\theta$y=sinθ have certain common properties. Each graph demonstrates repetition. We call the graphs of $y=\cos\theta$y=cosθ and $y=\sin\theta$y=sinθ periodic, or cyclic. We 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 behaviour, 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 $y=\sin\theta$y=sinθ 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. This means, the range of both the $\sin\theta$sinθ and $\cos\theta$cosθ functions is between $-1$1 and $1$1.

$y=\tan\theta$y=tanθ is also periodic, however when you look closely at its graph you can see that its cycle length is only $180^\circ$180°. Its range is unbounded, and it also has values of $\theta$θ for which the function cannot be calculated. This means that, unlike $\sin\theta$sinθ and $\cos\theta$cosθ, it is not defined for all real values.

Practice questions

QUESTION 1

Consider the curve $y=\sin x$y=sinx drawn below and answer the following questions.

Loading Graph...

  1. What is the $y$y-intercept? Give your answer as coordinates in the form $\left(a,b\right)$(a,b).

  2. What is the maximum $y$y-value?

  3. What is the minimum $y$y-value?

question 2

Consider the curve $y=\cos x$y=cosx drawn below and determine whether the following statements are true or false.

Loading Graph...

  1. The graph of $y=\cos x$y=cosx is cyclic.

    True

    A

    False

    B

    True

    A

    False

    B
  2. As $x$x approaches infinity, the height of the graph approaches infinity.

    True

    A

    False

    B

    True

    A

    False

    B
  3. The graph of $y=\cos x$y=cosx is increasing between $x=90^\circ$x=90° and $x=180^\circ$x=180°.

    True

    A

    False

    B

    True

    A

    False

    B

question 3

Given the unit circle, which of the following is true about the graph of $y=\tan x$y=tanx?

Select all that apply.

  1. 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.

    A

    The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.

    B

    Since the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$1y1.

    C

    The range of values of $y=\tan x$y=tanx is $-\infty<y<.

    D

    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.

    A

    The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.

    B

    Since the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$1y1.

    C

    The range of values of $y=\tan x$y=tanx is $-\infty<y<.

    D

Outcomes

0606C1.3

Understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic or trigonometric.

0606C10.2

Understand amplitude and periodicity and the relationship between graphs of related trigonometric functions, e.g. sin x and sin 2x.

0606C10.3

Draw and use the graphs of y = asinbx + c, y = acos bx + c, y = atan bx + c where a is a positive integer, b is a simple fraction or integer (fractions will have a denominator of 2, 3, 4, 6 or 8 only), and c is an integer.

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