5.05 Circular functions: sine and cosine

From our previous lessons we know that 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).

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?

Created with Geogebra

 

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:

 

$y=\sin\theta$y=sinθ

 

$y=\cos\theta$y=cosθ

 

Consequently, the graphs of $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ have many similar properties. Each graph demonstrates repetition. We call the graphs of $y=\sin\theta$y=sinθ and $y=\cos\theta$y=cosθ cyclical or periodic 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.

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 $y=\sin x$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 whether the $y$y value is negative or positive at this point. $\frac{24\pi}{12}$24π12​ is $2\pi$2π, therefore $\frac{23\pi}{12}$23π12​ is slightly to the left of $2\pi$2π.

Do: We plot the point on the graph of $y=\cos x$y=cosx below.

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 curve at this point is above the $x$x-axis, and therefore state that $\cos\frac{23\pi}{12}$cos23π12​ is positive.

Practice questions

question 1

QUESTION 2

question 3

question 4