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 rightangled 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 $y$yvalues of a point on the unit circle after having been rotated by an angle of measure $\theta$θ in an anticlockwise direction. Or, if $\theta$θ is negative, then the point is rotated in the clockwise direction.
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 $y$ycoordinate of the point on the unit circle.
The cosine of the angle is defined to be the $x$xcoordinate o the point on the unit circle.
The tangent of the angle can be defined algebraically as the ratio $\left(\frac{\sin\theta}{\cos\theta}\right)$(sinθcosθ). This also represents the gradient of the line that forms the angle $\theta$θ to the positive $x$xaxis. It can also be geometrically defined to be the $y$ycoordinate of a point $Q$Q, where $Q$Q is the intersection of the extension of the line $OP$OP and the tangent of the circle at $\left(1,0\right)$(1,0).
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.

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:
$y=\sin\theta$y=sinθ 
$y=\cos\theta$y=cosθ 
The simplest way to calculate $\tan\theta$tanθ is to use the values in the above graphs to evaluate $\frac{\sin\theta}{\cos\theta}$sinθcosθ, which gives us the following graph:
$y=\tan\theta$y=tanθ 
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 $y$yvalues 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°.
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 $y$yvalues increase as the $x$xvalues increase. Similarly, we say the graph is decreasing if the $y$yvalues decrease as the $x$xvalues increase.
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.
$y=\tan\theta$y=tanθ is also periodic, however when you look closely at its graph you can see that its cycle length is only $180^\circ$180°. Its range is unbounded, and it also has values of $\theta$θ for which the function cannot be calculated. This means that, unlike $\sin\theta$sinθ and $\cos\theta$cosθ, it is not defined for all real values.
Consider the curve $y=\sin x$y=sinx drawn below and determine whether the following statements are true or false.
The graph of $y=\sin x$y=sinx is periodic.
True
False
True
False
As $x$x approaches infinity, the height of the graph approaches infinity.
True
False
True
False
The graph of $y=\sin x$y=sinx is increasing between $x=\left(90\right)^\circ$x=(−90)° and $x=0^\circ$x=0°.
False
True
False
True
Consider the graph of $y=\tan x$y=tanx given below.
Using the graph, what is the sign of $\tan340^\circ$tan340°?
Negative
Positive
Negative
Positive
Which quadrant does an angle with measure $340^\circ$340° lie in?
First quadrant
Second quadrant
Third quadrant
Fourth quadrant
First quadrant
Second quadrant
Third quadrant
Fourth quadrant
Consider the curve $y=\sin x$y=sinx drawn below and answer the following questions.
What is the $x$xvalue of the $x$xintercept in the region $\left(360\right)^\circ
Determine whether the following statement is true or false.
As $x$x approaches infinity, the graph of $y=\sin x$y=sinx stays between $y=1$y=−1 and $y=1$y=1.
True
False
True
False
Which of the following regions is the graph of $y=\sin x$y=sinx increasing? Select all that apply.
$90^\circ
$\left(270\right)^\circ
$270^\circ
$\left(450\right)^\circ
$90^\circ
$\left(270\right)^\circ
$270^\circ
$\left(450\right)^\circ
Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies.