3.01 Sine and cosine functions

 

Before looking at applications and calculus involving trigonometric functions, let's review the functions of sine, and cosine.

 

$\sin\left(\theta\right)$sin(θ) and $\cos\left(\theta\right)$cos(θ)

We will begin by looking at the unit circle, that is, a circle with the origin at the centre and a unit radius (radius equals one). 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).

The unit circle

 

Exact values of the trigonometric ratios exist within both the unit circle and the trigonometric functions.  The basic ratios in Quadrant 1 are as follows:

	$\sin\theta$sinθ	$\cos\theta$cosθ	$\tan\theta$tanθ
$\frac{\pi}{6}$π6​	$\frac{1}{2}$12​	$\frac{\sqrt{3}}{2}$√32​	$\frac{1}{\sqrt{3}}$1√3​
$\frac{\pi}{4}$π4​	$\frac{1}{\sqrt{2}}$1√2​	$\frac{1}{\sqrt{2}}$1√2​	$1$1
$\frac{\pi}{3}$π3​	$\frac{\sqrt{3}}{2}$√32​	$\frac{1}{2}$12​	$\sqrt{3}$√3

In order to use these Quadrant 1 ratios to find those in the other Quadrants, the following must be noted:

Quadrant	Domain	Rule	Sign of Solutions
I	$0\le x\le\frac{\pi}{2}$0≤x≤π2​	$\theta$θ	All three functions are positive
II	$\frac{\pi}{2}\le x\le\pi$π2​≤x≤π	$\pi-\theta$π−θ	Cosine and Tangent are negative Sine is positive
III	$\pi\le x\le\frac{3\pi}{2}$π≤x≤3π2​	$\pi+\theta$π+θ	Cosine and Sine are negative Tangent is positive
IV	$\frac{3\pi}{2}\le x\le2\pi$3π2​≤x≤2π	$2\pi-\theta$2π−θ	Sine and Tangent are negative Cosine is positive

Worked Example

Example 1

Find the exact value of $\cos(\frac{2\pi}{3}).$cos(2π3​).

$\frac{2\pi}{3}$2π3​ is in the second quadrant.  Therefore, we should think of $\frac{2\pi}{3}$2π3​  as $\pi-\frac{\pi}{3}.$π−π3​.  As we know that $\cos(\frac{\pi}{3})=\frac{1}{2}$cos(π3​)=12​  and the cosine function is negative in the second quadrant, the solution is as follows:

$\cos(\frac{2\pi}{3})$cos(2π3​)	$=$=	$-\cos(\frac{\pi}{3})$−cos(π3​)
	$=$=	$-\frac{1}{2}$−12​

      

If we imagine the point moving around the unit circle, 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 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.

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

Example 2

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.

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

question 2

question 3