27.02 Graphs of sine and cosine functions (Extended)

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°≤θ≤90°.

 

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.

 

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.

Created with Geogebra

Graphs of sin and cos 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

 

Properties of trigonometric graphs

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. 

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.

 

Practice questions

QUESTION 1

 

question 2