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
pi/6 1/2 SQRT(3)/2 SQRT(3)/3 2 2SQRT(3)/3 SQRT(3)
pi/4 SQRT(2)/2 SQRT(2)/2 1 SQRT(2) SQRT(2) 1
pi/3 SQRT(3)/2 1/2 SQRT(3) 2SQRT(3)/3 2 SQRT(3)/3
pi/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 = npi (where n is any integer). In the functions tant .= y/x and sect..= 1/x, then x not equal to 0. This happens when t..= pi/2 + npi (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:

sin, cosall real numbers
tan, secall real number other than
pi/2 + npi for any integer n
cot, cscall real numbers other than
npi 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

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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