Expert: Josh - 2/23/2005

Question
If tan0=radical11/5 and secant0>0, find
a)sec0 b)cos0 c)sin0 d)csc0 e)cot0

Answer
Understand this. All these fancy terms used in trigonometry are simply the ratios between the lengths of two sides in a right-angle triangle.

All you really need to remember are three definitions.
======================TABLE 1=========================
SINE is the ratio between opposite side and the hypotenuse
COSINE is the ratio betw. the adjacent side and the hypotenuse.
TANGENT is the ratio betw. the opposite side and the adjacent side.
======================================================

We will use "x" instead of "theta" to refer to the angle ^ABC in the following diagram. (Draw a line from A to B)

A
|
|
|
|         x
C-------------B
Figure 1: a right angle triangle.

Clarification:
As you can see |AC| is directly opposite the angle x.
|AB| is called the hypotenuse, because it is the longest side in the right angle triangle. Finally, we refer to side |BC| as the adjacent side, in relation to angle x.

Using those definitions,
sin(x)= |AC|/|AB|,
cos(x)= |BC|/|AB|,
tan(x)= |AC|/|BC|.

Now, secant(x) is simply the inverse of cos(x).
Similarly, cosecant(x) is simply the inverse of sin(x).
You guessed it, cotangent(x) is the inverse of tan(x).

So, its not worth putting more into your brain.
They follow from the definitions of sine, cos and tan.

Let me spell this out more clearly,
======================TABLE 2=========================
COSECANT (CSC) is the length of the hypotenuse, divided by the length of the opposite side.
SECANT (SEC) is the length of the hypotenuse, divided by the length of the adjacent side.
COTANGENT (COT) is the length of the adjacent side, divided by the length of the opposite side.

======================================================
sin(x)= |AC|/|AB|, cosecant(x)=|AB|/|AC|,
cos(x)= |BC|/|AB|, secant(x)=|AB|/|BC|,
tan(x)= |AC|/|BC|, cotangent(x)=|BC|/|AC|.
======================TABLE 3=========================

Using these, you can solve all similar problems yourself. Here's how.
You were told that tan(x)=11/5. Referring to figure 1,
the opposite side to angle x is |AC|, its length is 11.
The adjacent side to angle x is |BC|, its length is 5.
Using Pythagoras theorem, |AB|=square_root(|AC|*|AC|+|BC|*|BC|)=sqrt(121+25)=sqrt(146)

You can work out all the other ratios using Table 3.
For instance, cosecant is nothing more than the inverse of sine. Sine is the (opposite side over hypotenuse). So, cosecant is the (hypotenuse over the opposite side).
Referring to the diagram, csc(x)=|AB|/|AC|. You can plug in the numbers. Just remember the three basic definitions and you'll be fine.

Cheers.