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:

t	sint	cost	tant	csct	sect	cott
0	0	1	0	undefined	1	undefined
/6	1/2	/2	/3	2	2/3
/4	/2	/2	1			1
/3	/2	1/2		2/3	2	/3
/2	1	0	undefined	1	undefined	0

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 y = 0. Hence, we cannot have t = n (where n is any integer). In the functions tant .= y/x and sect..= 1/x, then x 0. This happens when t..= /2 + n (where n is any integer). Here are the graphs of what the trigonometric functions look like.

Now condensing this information into a table we get:

Function	Domain
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.

Note:

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

The only even functions are cosine and secant. The rest are odd functions.

Now using the tables and the diagram, lets do a few examples.

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