When looking at the unit circle, that is a circle with the origin at the centre and unit radius, the coordinates of any point on that circle can be described using trigonometry. Specifically a point on the circle at an angle of $\theta$θ anticlockwise from the $x$x-axis has coordinates $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ).
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| The unit circle |
Let's revisit the applet from our introduction to the unit circle. Start with an angle of $0^\circ$0° and slowly increase the angle. If we were to graph the height of the blue line ($\sin\theta$sinθ) above the $x$x-axis, what value would it start at? What values would it be bound between? If any angle was allowed, how often would the pattern repeat? What would be different about graphing the distance the red line ($\cos\theta$cosθ) reaches in front of the $y-axis$y−axis?
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As we move through different values of $\theta$θ the value of $\sin\theta$sinθ and $\cos\theta$cosθ move accordingly between $-1$−1 and $1$1. If we plot the values of $\sin\theta$sinθ and $\cos\theta$cosθ according 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 |
Consequently, the graphs of $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ have many properties. Each graph demonstrates repetition. We call the graphs of $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ cyclical 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 $2\pi$2π.
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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 $y=\sin x$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.
Worked example
By using the graph of $y=\cos x$y=cosx, what is the sign of $\cos\frac{23\pi}{12}$cos23π12?
Think: Using the graph of $y=\cos x$y=cosx, we can roughly estimate where the point $\left(\frac{23\pi}{12},\cos\frac{23\pi}{12}\right)$(23π12,cos23π12) lies and from this, determine the sign of $\cos\frac{23\pi}{12}$cos23π12.
Do: We plot the point on the graph of $y=\cos x$y=cosx below.
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The point $\left(\frac{23\pi}{12},\cos\frac{23\pi}{12}\right)$(23π12,cos23π12) drawn on the graph of $y=\cos x$y=cosx. |
We can quickly observe that the height of the curve at this point is above the $x$x-axis, and observe that $\cos\frac{23\pi}{12}$cos23π12 is positive.
Practice questions
question 1
Consider the equation $y=\sin x$y=sinx.
Using the fact that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$sinπ3=√32, what is the value of $\sin\frac{2\pi}{3}$sin2π3?
Using the fact that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$sinπ3=√32, what is the value of $\sin\frac{4\pi}{3}$sin4π3?
Using the fact that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$sinπ3=√32, what is the value of $\sin\frac{5\pi}{3}$sin5π3?
Complete the table of values. Give your answers in exact form.
$x$x $0$0 $\frac{\pi}{3}$π3 $\frac{\pi}{2}$π2 $\frac{2\pi}{3}$2π3 $\pi$π $\frac{4\pi}{3}$4π3 $\frac{3\pi}{2}$3π2 $\frac{5\pi}{3}$5π3 $2\pi$2π $\sin x$sinx $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Draw the graph of $y=\sin x$y=sinx.
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QUESTION 2
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 3
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 for $y=\cos x$y=cosx approaches infinity.
True
AFalse
BThe graph of $y=\cos x$y=cosx is increasing between $x=-\frac{\pi}{2}$x=−π2 and $x=0$x=0.
False
ATrue
B
question 4
Consider the curve $y=\sin x$y=sinx drawn below and answer the following questions.
If one cycle of the graph of $y=\sin x$y=sinx starts at $x=0$x=0, when does the next cycle start?
In which of the following regions is the graph of $y=\sin x$y=sinx decreasing? Select all that apply.
$-\frac{\pi}{2}
A$\frac{\pi}{2}
B$-\frac{5\pi}{2}
C$-\frac{3\pi}{2}
DWhat is the $x$x-value of the $x$x-intercept in the region $0