6.05 Trigonometric functions
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.
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| 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.
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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:
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| $y=\sin\theta$y=sinθ |
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| $y=\cos\theta$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°.
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| 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.
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| 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
Consider the equation $y=\cos x$y=cosx.
Complete the table of values.
Give your answers in exact form.
$x$x $0^\circ$0° $90^\circ$90° $180^\circ$180° $270^\circ$270° $360^\circ$360° $\cos x$cosx $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the graph of $y=\cos x$y=cosx.
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question 2
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 $x=-180^\circ$x=−180° and $x=-90^\circ$x=−90°.
True
AFalse
B
question 3
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 4
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 $x=-90^\circ$x=−90°, when does the next cycle start?
In which two of the following regions is the graph of $y=\cos x$y=cosx decreasing?
$180^\circ
A$-360^\circ
B$-180^\circ
C$0^\circ
DWhat are the $x$x-values of the $x$x-intercepts in the region $0^\circ