Activities
Trigonometry - definitions of cosine, sine and tangent for any angle
In order to use the full power of trigonometry, it is necessary to define cosine, sine and tangent for any angle, not just positive acute angles.
Let the hypotenuse of the right angled triangle have a length of 1 unit. Then the lengths of the other sides are the values of cosine and sine.
Place the triangle on the cartesian plane so that the angle is measured from the x-axis.
Draw the unit circle. The coordinates of point P are the values of the cosine and sine of the angle.
As the angle changes, P moves around the unit circle.
The new definitions for the trigonometric ratios are:
The connection between the definitions of cosine, sine and tangent and their graphs is explored in the java applets.
An interactive tutorial is available that deals with trigonometric ratios for any angle.
See also Dave's Short Trig Course.
The values of cosine, sine and tangent can also be explored by investigating their graphs.
When is cosine positive and when is it negative? What is the connection with the unit circle definitions?
When is sine positive and when is it negative? What is the connection with the unit circle definitions?
When is tangent positive and when is it negative? What is the connection with the unit circle definitions?
Investigate moving the cosine graph 90° to the right.
Explain what happens by referring to the unit circle definitions.
Investigate moving the sine graph 90° to the right?
The tangent graph is discontinuous (there are breaks) and is better drawn with MODE Dot instead of MODE Connected.
Why does tan 90° not exist?
The use of MODE Connected involves the calculator marking a sequence of points and then connecting them in pairs by straight lines. Some of these lines cross over the breaks in the graph.
The following special results apply.
In particular, consider the case of the obtuse angle 150° (animation).
