More Trig. Functions
We know that from the previous sections (Trig. Functions and circles) there are special reference angles in the unit circle that come up in many of the calculations. Here is a quick reference of the trig values of these angles:
From the table we see that some of the trigonometric functions are not defined
for certain real numbers. Therefore, we must determine their domains. The
functions sin and cos are defined for all values of t . Since cotangent
and cosecant are defined by x/y and 1/y, respectively, we cannot have
Now condensing this information into a table we get:
|sin, cos||all real numbers|
|tan, sec||all real number other than|
/2 + n for any integer n
|cot, csc||all real numbers other than|
n for any integer n
Let's take a look at a diagram to see the signs of the trig. functions in each quadrant.
We can find the even-odd properties of the trigonometric functions using the diagram above along with this circle.
- An odd function f is one that satisifies
f(-x) = -f(x)for all x in its domain.
An even function f is one that satisifies
f(-x) = f(x)for all x in its domain.
We see that:
- sin(-t) = -y = -sint________csc(-t) = -1/y = -csct
cos(-t) = x = cost________--sec(-t) = 1/x = sect
tan(-t) = -y/x = -tant_______cot(-t) = x/(-y) = -cott
Now using the tables and the diagram, lets do a few examples.