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7.02 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 is a circle of radius equal to $1$1 centred at the origin. By looking at the trigonometric ratios, we get the definitions of $\sin\theta$sinθ and $\cos\theta$cosθ on the unit circle as the $x$x and $y$y-values of a point on the unit circle after having been rotated by an angle of measure $\theta$θ in an anti-clockwise 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°.

 

The sine of the angle is defined to be the $y$y-coordinate of the point on the unit circle.

The cosine of the angle is defined to be the $x$x-coordinate o the point on the unit circle.

The tangent of the angle can be defined algebraically as the ratio $\left(\frac{\sin\theta}{\cos\theta}\right)$(sinθcosθ). This also represents the gradient of the line that forms the angle $\theta$θ to the positive $x$x-axis. It can also be geometrically defined to be the $y$y-coordinate of a point $Q$Q, where $Q$Q is the intersection of the extension of the line $OP$OP and the tangent of the circle at $\left(1,0\right)$(1,0).

 

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 where $\cos\theta$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 determine whether the following statements are true or false.

Loading Graph...

  1. The graph of $y=\sin x$y=sinx is periodic.

    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 $y=\sin x$y=sinx is increasing between $x=\left(-90\right)^\circ$x=(90)° and $x=0^\circ$x=0°.

    False

    A

    True

    B

    False

    A

    True

    B

question 2

Consider the graph of $y=\tan x$y=tanx given below.

Loading Graph...

  1. Using the graph, what is the sign of $\tan340^\circ$tan340°?

    Negative

    A

    Positive

    B

    Negative

    A

    Positive

    B
  2. Which quadrant does an angle with measure $340^\circ$340° lie in?

    First quadrant

    A

    Second quadrant

    B

    Third quadrant

    C

    Fourth quadrant

    D

    First quadrant

    A

    Second quadrant

    B

    Third quadrant

    C

    Fourth quadrant

    D

question 3

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

Loading Graph...

  1. What is the $x$x-value of the $x$x-intercept in the region $\left(-360\right)^\circ(360)°<x<0°?

  2. Determine whether the following statement is true or false.

    As $x$x approaches infinity, the graph of $y=\sin x$y=sinx stays between $y=-1$y=1 and $y=1$y=1.

    True

    A

    False

    B

    True

    A

    False

    B
  3. Which of the following regions is the graph of $y=\sin x$y=sinx increasing? Select all that apply.

    $90^\circ90°<x<270°

    A

    $\left(-270\right)^\circ(270)°<x<(90)°

    B

    $270^\circ270°<x<450°

    C

    $\left(-450\right)^\circ(450)°<x<(270)°

    D

    $90^\circ90°<x<270°

    A

    $\left(-270\right)^\circ(270)°<x<(90)°

    B

    $270^\circ270°<x<450°

    C

    $\left(-450\right)^\circ(450)°<x<(270)°

    D

Outcomes

VCMMG368 (10a)

Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies.

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