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