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Reciprocal Trigonometric Functions
Introduction: In this lesson, three trigonometric ratios (secant, cosecant and cotangent) will be defined and applied. These involve ratios of the lengths of the sides in a right triangle.

The Lesson:
In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. The two sides which form the 90º angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.

We shall call the opposite side “opp,” the adjacent side “adj” and the hypotenuse “hyp.”
Definitions:
In the following definitions, sine is called “sin,” cosine is called “cos” and tangent is called “tan.” The origin of these terms relates to arcs and tangents to a circle.
  1. sin(A) =
  2. cos(A) =
  3. tan(A) =
For more information, reference the lesson on sine, cosine, and tangent.

Three more trigonometric ratios can be defined as the reciprocals of these fundamental ratios. They are cosecant, secant, and cotangent. The ratios are given by the following equations:
  1. csc(A) =
  2. sec(A) =
  3. cot(A) =
For example, in the triangle shown below, with respect to angle q, opp = 5, hyp = 6, and by the Pythagorean Theorem, adj = .

By the three definitions we have:
csc(q) = sec(q) = cot(q) =
Notes:
Often, fractions involving radicals are rewritten so that there is not a radical expression in the denominator.
Rationalizing the demoniator we would get sec(q) = .


We can use any of these three reciprocal trig ratios to find the measure of an unknown angle q.

For example, if sec(q) = 1.7, we can use a calculator to determine what angle has this secant. Using a TI-83 calculator, we press [(1.7) ] and get . We use the key because cosine is the reciprocal of secant and we need to invert the value 1.7 and use the key since the calculator does not have a secant key.
Identities:
It is not as easy to find angle and side measurements of triangles using the reciprocal functions as it is to use sine, cosine, and tangent. This is primarily because the calculators typically have sin, cos, and tan keys and not keys for the reciprocal functions. However, we can use these reciprocal functions to provide identities which are equations relating trig functions to each other.

An example of an identity with the variable x is
2x(3 – x) = 6x – 2x2.

This statement is true for ALL values of x.
An identity involving trig functions is
.
This statement is true for any angle A.
A second example of a trig identity is
tan(A)csc(A) = sec(A).

To show that this statement is true, substitute and simplify.

tan(A)csc(A) = .
Let's Practice:
  1. In the triangle below, what is the csc(q) and what is the length of the adjacent side? What is the measure of angle q?

The adjacent side can be found using the Pythagorean Theorem and is .

We can determine the measure of q by working with
csc(q) = .
Using we get . Remember, we needed to use 2nd sin because there is no csc key on the calculator.

We could also use a trig function involving the adjacent side. Using we would still get .
  1. Show that cot(A)sin(A) = cos(A).
Use to substitute and simplify.

cot(A)sin(A) =

Examples
Example In a right triangle, the legs are 6 and . What is the secant of the angle opposite the side of measure 6? What is the cotangent of this angle?
What is your answer?
 
Example Show that cot(A)sec(A) = csc(A).
What is your answer?
 



M Ransom

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