In Geometry Trigonometric Functions we saw that there are 3 basic trigonometric ratios. We will now be adding the reciprocals of those ratios to create a total of 6 trigonometric ratios.

 How many trig ratios?

So, how many ratios pertaining to the sides of the triangle are possible? Let's take a look:
What possible ratios
of the sides exist?

If we flip these three ratios over, we have three more:

 Naming the 6 ratios:

The first three ratios established above have specific "names" (sine, cosine and tangent). These are referred to as the basic trigonometric functions.

 Sine (sin) Cosine (cos) Tangent (tan)

The second three ratios established above also have specific "names" (cosecant, secant, and cotangent). These three ratios are referred to as the reciprocal trigonometric functions.

 Cosecant (csc) Secant (sec) Cotangent (cot)

Notice that these three new ratios are reciprocals of the ratios of the basic trigonometric functions.
Applying a little algebra shows the connections between these functions.

Applying this connection will create some basically used statements about trigonometric ratios:

 Reciprocal Functions
 Relationships

 Trigonometric functions work ONLY in right triangles!

 Given the triangle shown at the right, express the exact value of the six trigonometric 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.  Since the question asks for the "exact" value, do not "estimate" the answers. Be careful not to jump to the conclusion that this is a 3-4-5 right triangle.  The 4 in on the hypotenuse and must be the largest side.

 Assume 0º < θ < 90º. Solution:  This is an easy problem to solve. Since cosine and secant are reciprocal functions, simply invert (or take the reciprocal of ) 12/13.

Assume 0º < θ < 90º.
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: