Trig Function Point Definitions
See Quick Instructions and Further Discussion below...
- The above applet starts with the sine function active. Any of the six trigonometric functions
can be activated by choosing the appropriate radio button at the top of the applet.
- The large square graph on the left is the (x, y) coordinate plane. It extends from -10.0 to
+10.0 along the x-axis and the y-axis.
- The upper right rectangular graph shows the graph of the currently active trig function. For
example, when the sine function is active, this graph shows the sin(a) vs. a, where a represents the angle. Horizontally,
this graph has a domain from 0 radians to 2pi radians with markers every pi/2 radians. Vertically, the range of
this graph runs from -2.0 to +2.0 for the sine and cosine functions, and it runs from -5.0 to +5.0 for the tangent,
cotangent, secant and cosecant functions. Vertical marking occur every unit distance.
- The lower right rectangular area shows the calculation for the currently active trig function
at the current (x, y) point on the left graph.
- When the sine function radio button is selected, click somewhere on the left (x, y) graph. Notice:
- The (x, y) coordinates are presented above the selected point.
- The angle whose terminal side goes through this (x, y) point is drawn. Its value is presented
in the upper right corner of this (x, y) graph.
- The relevant quantities necessary for the sine calculation are drawn on the graph, (the y distance
and the radius in this case). The values for these quantities are presented near their graphic representations.
- The upper right hand graph shows the current function, i. e., the sine function. The current
input angle and the current output value for this function are presented.
- In the lower right rectangle of the applet the value for the current trig function at the current
angle is calculated using the the relevant quantities from the (x, y) graph.
- All values are rounded to two decimal places. Certainly more precise values for the trig functions
are available elsewhere. This applet, though, is not meant to be a calculator. It is meant to demonstrate the interrelationships
of several trigonometric concepts.
- Try different (x, y) positions and different trig functions.
This material explains the definitions of the six trigonometric functions in terms of an (x,
y) point located on the terminal side of the input angle. You should be familiar with:
- The (x, y) coordinate plane.
- How to find the distance from the origin to an (x, y) point.
- The graphs of the trig functions.
- Angles in standard position.
- Radian measure for angles.
The six trigonometric functions, (sine, cosine, tangent, cotangent, secant and cosecant), are
usually thought to accept an angle as input and output a pure number. For the purposes of the definitions this
angle is to be placed in standard position. We will be concerned with any (x, y) point located on the terminal
side of this angle. These definitions are based on such an (x, y) point.
These definitions also use the distance from the origin to the (x, y) point. This distance will
be referred to as r and can be calculated like this:
The six definitions are:
sin(angle) = y/r
cos(angle) = x/r
tan(angle) = y/x (x not equal to zero)
csc(angle) = r/y (y not equal to zero)
sec(angle) = r/x (x not equal to zero)
cot(angle) = x/y (y not equal to zero)
So, for example, the point (5, 7) is on the terminal side of an angle in standard position which
has a measure of about 0.95 radians (about 54 degrees):
The distance from the origin to the point (5, 7) can be calculated this way (approximate result
Therefore, the six trigonometric function values for this angle can be calculated as follows
(approximate results given):
|sin(0.95) = y/r = 7/8.6 = 0.81
|cos(0.95) = x/r = 5/8.6 = 0.58
|tan(0.95) = y/x = 7/5 = 1.4
|csc(0.95) = r/y = 8.6/7 = 1.2
|sec(0.95) = r/x = 8.6/5 = 1.7
|cot(0.95) = x/y = 5/7 = 0.71
In all of the above calculations approximate results were given. Of course, if you are doing
more careful and important work you would use measurements and calculations of higher significance.
The definitions given here for the six trigonometric functions are more powerful than the right
triangle definitions given in the introduction to trigonometry section. The right triangle definitions are only
good for angles up to pi/2 radians (90 degrees). These trig definitions based upon an (x, y) point on the terminal
side of an angle are good for angles of any measurement, positive or negative.