In a
right triangle,
there are actually six possible trigonometric ratios, or
functions.
A Greek letter (such as theta
or
phi )
will now be used to represent the angle.
Notice that the three
new ratios at the right are reciprocals of the ratios on
the left.
Applying a little algebra shows the connection between these functions.
Examples:
1.
Given the triangle at the right, express
the exact value of the six trig functions in
relation to theta. 

Solution:
Find the missing side of the right triangle
using the Pythagorean Theorem. Then, using
the diagram, express each function as a ratio of
the lengths of the sides. Do not
"estimate" the answers.


Be careful not to jump
to the conclusion that this is a 345
right triangle. The 4 in on the
hypotenuse and must be the largest side.



The following examples pertain to a right triangle in Quadrant I:
2.
Given 

, find
.

Solution:
This is an easy problem, since cosine and secant
are reciprocal functions.

3.
Find
and
,
given 
and
.

Solution:
Draw a diagram to get a better understanding of
the given information.

Since
sine is opposite over hypotenuse,
position the 2 and the 3 accordingly in
relation to the angle theta. Now,
since cosine is adjacent over hypotenuse,
position these values (the 3 should
already be properly placed). Be
sure that the largest value is on the
hypotenuse and that the Pythagorean
Theorem is true for these values.
(If you are not given the third side,
use the Pythagorean Theorem to find it.) 
Now, using your diagram,
read off the values for the secant and
the cotangent.
Secant:

Cotangent:




Check out how to use your
TI83+/84+ graphing calculator with
reciprocal functions.
Click here. 

