Unit Circle and the Trigonometric Functions sin(x), cos(x) and tan(x)

Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions.

The relationships between the graphs (in rectangular coordinates) of sin(x), cos(x) and tan(x) and the coordinates of a point on a unit circle are explored using an applet.


1- Let x be a real number and P(x) a point on a unit circle such that the angle in standard position whose terminal side is segment OP is equal to x radians.(O is the origin of the system of axis used).

2- We define sin(x) as the y-coordinate of point P(x) on the unit circle.

3- We define cos(x) as the x-coordinate of a point P(x) on the unit circle.

4- We define tan(x) as the ratio of the y-coordinate and x-coordinate of point P(x) on a unit circle.

Interactive Tutorial Using Applets

Two possibilities to explore trigonometric function using the unit circle.

1) Using a Unit Circle HTML5 Applet
A java applet below
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  1. Is there a point P(x) that cannot have any values for its x or y-coordinates? The x and y-coordinates are cos(x) and sin(x), what is the domain of sin(x), what is the domain of cos(x)?

  2. Explore the x-intercepts, the maximums and minimums (if any) of the graphs of sin(x) and cos(x) using the unit circle.

  3. Using the unit circle, do you think that any of the coordinates of a point on the circle can be larger than 1 or smaller than -1. Why do you think that sin(x) and cos(x) cannot be larger than 1 or smaller than -1?

  4. Explore the periodicity of sin(x), cos(x) and tan(x).

  5. tan(x)is the ratio y-coordinate / x-coordinate and whenever the x-coordinate of point P(x) is equal to zero, we cannot define tan(x).Find these points for x between zero and 2Pi. What is the domain of tan(x)?

  6. At these same points where tan(x)is undefined, the graph of tan(x) on the right shows an asymptotic behavior, Explain. What do you think is the range of tan(x)? Find all x-intercepts of tan(x) between 0 and 2Pi (inclusive). Explain their positions on the x-axis.

More references on unit circle and trigonometric functions.