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Trigonometric Functions and the Unit Circle
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Concept Version 4
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Defining Trigonometric Functions on the Unit Circle

Identifying points on a unit circle allows one to apply trigonometric functions to any angle.

Learning Objective

  • Use right triangles drawn in the unit circle to define the trigonometric functions for any angle


Key Points

    • The xxx- and yyy-coordinates at a point on the unit circle given by an angle ttt are defined by the functions x=costx = \cos tx=cost and y=sinty = \sin ty=sint.
    • Although the tangent function is not indicated by the unit circle, we can apply the formula tant=sintcost\displaystyle{\tan t = \frac{\sin t}{\cos t}}tant=​cost​​sint​​ to find the tangent of any angle identified.
    • Using the unit circle, we are able to apply trigonometric functions to any angle, including those greater than 90∘90^{\circ}90​∘​​.
    • The unit circle demonstrates the periodicity of trigonometric functions by showing that they result in a repeated set of values at regular intervals.

Terms

  • periodicity

    The quality of a function with a repeated set of values at regular intervals.

  • unit circle

    A circle centered at the origin with radius 1.

  • quadrants

    The four quarters of a coordinate plane, formed by the xxx- and yyy-axes.


Full Text

Trigonometric Functions and the Unit Circle

We have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in terms of the unit circle. Recall that a unit circle is a circle centered at the origin with radius 1. The angle ttt (in radians) forms an arc of length sss.

The x- and y-axes divide the coordinate plane (and the unit circle, since it is centered at the origin) into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.

For any angle ttt, we can label the intersection of its side and the unit circle by its coordinates, (x,y)(x, y)(x,y). The coordinates xxx and yyy will be the outputs of the trigonometric functions f(t)=costf(t) = \cos tf(t)=cost and f(t)=sintf(t) = \sin tf(t)=sint, respectively. This means:

$\displaystyle{ \begin{aligned} x &= \cos t \\ ​y &= \sin t \end{aligned} }$ 

The diagram of the unit circle illustrates these coordinates.  

Unit circle

Coordinates of a point on a unit circle where the central angle is ttt radians.

Note that the values of xxx and yyy are given by the lengths of the two triangle legs that are colored red. This is a right triangle, and you can see how the lengths of these two sides (and the values of xxx and yyy) are given by trigonometric functions of ttt.

For an example of how this applies, consider the diagram showing the point with coordinates (−22,22)\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }(−​2​​√​2​​​​​,​2​​√​2​​​​​) on a unit circle.

Point on a unit circle

The point (−22,22)\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }(−​2​​√​2​​​​​,​2​​√​2​​​​​) on a unit circle.

We know that, for any point on a unit circle, the xxx-coordinate is cost\cos tcost and the yyy-coordinate is sint\sin tsint. Applying this, we can identify that cost=−22\displaystyle{ \cos t = -\frac{\sqrt2}{2} }cost=−​2​​√​2​​​​​ and  sint=−22\displaystyle{\sin t = -\frac{\sqrt2}{2}}sint=−​2​​√​2​​​​​ for the angle ttt in the diagram. 

Recall that tant=sintcost\displaystyle{\tan t = \frac{\sin t}{\cos t}}tant=​cost​​sint​​.  Applying this formula, we can find the tangent of any angle identified by a unit circle as well. For the angle ttt identified in the diagram  of the unit circle showing the point (−22,22)\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }(−​2​​√​2​​​​​,​2​​√​2​​​​​), the tangent is:

tant=sintcost=−22−22=1\displaystyle{ \begin{aligned} \tan t &= \frac{\sin t}{\cos t} \\ &= \frac{-\frac{\sqrt2}{2}}{-\frac{\sqrt2}{2}} \\ &= 1 \end{aligned} }​tant​​​​​=​cost​​sint​​​=​−​2​​√​2​​​​​​​−​2​​√​2​​​​​​​​=1​​

We have previously discussed trigonometric functions as they apply to right triangles. This allowed us to make observations about the angles and sides of right triangles, but these observations were limited to angles with measures less than 90∘90^{\circ}90​∘​​. Using the unit circle, we are able to apply trigonometric functions to angles greater than 90∘90^{\circ}90​∘​​.

Further Consideration of the Unit Circle

The coordinates of certain points on the unit circle and the the measure of each angle in radians and degrees are shown in the unit circle coordinates diagram. This diagram allows one to make observations about each of these angles using trigonometric functions.

Unit circle coordinates

The unit circle, showing coordinates and angle measures of certain points.

We can find the coordinates of any point on the unit circle. Given any angle ttt, we can find the xxx- or yyy-coordinate at that point using x=cos tx = \text{cos } tx=cos t  and y=sin ty = \text{sin } ty=sin t.

The unit circle demonstrates the periodicity of trigonometric functions. Periodicity refers to the way trigonometric functions result in a repeated set of values at regular intervals. Take a look at the xxx-values of the coordinates in the unit circle above for values of ttt from 000 to 2π2{\pi}2π:

1,32,22,12,0,−12,−22,−32,−1,−32,−22,−12,0,12,22,32,1{1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0, -\frac{1}{2}, -\frac{\sqrt{2}}{2}, -\frac{\sqrt{3}}{2}, -1, -\frac{\sqrt{3}}{2}, -\frac{\sqrt{2}}{2}, -\frac{1}{2}, 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1}1,​2​​√​3​​​​​,​2​​√​2​​​​​,​2​​1​​,0,−​2​​1​​,−​2​​√​2​​​​​,−​2​​√​3​​​​​,−1,−​2​​√​3​​​​​,−​2​​√​2​​​​​,−​2​​1​​,0,​2​​1​​,​2​​√​2​​​​​,​2​​√​3​​​​​,1

We can identify a pattern in these numbers, which fluctuate between −1-1−1 and 111. Note that this pattern will repeat for higher values of ttt. Recall that these xxx-values correspond to cost\cos tcost. This is an indication of the periodicity of the cosine function.

Example 

Solve sin(7π6)\displaystyle{ \sin{ \left(\frac{7\pi}{6}\right) } }sin(​6​​7π​​). 

It seems like this would be complicated to work out. However, notice that the unit circle diagram shows the coordinates at t=7π6\displaystyle{ t = \frac{7\pi}{6} }t=​6​​7π​​. Since the yyy-coordinate corresponds to sint\sin tsint, we can identify that 

sin(7π6)=−12\displaystyle{\sin{ \left(\frac{7\pi}{6}\right)} = -\frac{1}{2} }sin(​6​​7π​​)=−​2​​1​​

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