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3.1.3: Reciprocal Identities

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Relationship between sine / cosine / tangent and cosecant / secant / cotangent.

You are already familiar with the trig identities of sine, cosine, and tangent. As you know, any fraction also has an inverse, which is found by reversing the positions of the numerator and denominator.

Can you list what the ratios would be for the three trig functions (sine, cosine, and tangent) with the numerators and denominators reversed?

Reciprocal Identities

A reciprocal of a fraction ab is the fraction ba. That is, we find the reciprocal of a fraction by interchanging the numerator and the denominator, or flipping the fraction. The six trig functions can be grouped in pairs as reciprocals.

First, consider the definition of the sine function for angles of rotation: sinθ=yr. Now consider the cosecant function: cscθ=ry. In the unit circle, these values are sinθ=y1=y and cscθ=1y. These two functions, by definition, are reciprocals. Therefore the sine value of an angle is always the reciprocal of the cosecant value, and vice versa. For example, if sinθ=12, then cscθ=21=2.

Analogously, the cosine function and the secant function are reciprocals, and the tangent and cotangent function are reciprocals:

secθ=1cosθ or cosθ=1secθcotθ=1tanθ or tanθ=1cotθ

Using Reciprocal Identities

Find the value of the following expressions using a reciprocal identity.

1. cosθ=.3, secθ=?

secθ=103

These functions are reciprocals, so if cosθ=.3, then secθ=1.3. It is easier to find the reciprocal if we express the values as fractions: cosθ=.3=310secθ=103.

2. cotθ=43, tanθ=?

These functions are reciprocals, and the reciprocal of 43 is 34.

We can also use the reciprocal relationships to determine the domain and range of functions.

3. sinθ=12, cscθ=?

These functions are reciprocals, and the reciprocal of 12 is 2.

Example 3.1.3.1

Earlier, you were asked to list the ratios for the three trigonometric functions with the numerators and denominators reversed.

Solution

Since the three regular trig functions are defined as:

sin=oppositehypotenusecos=adjacenthypotenusetan=oppositeadjacent

then the three functions - called "reciprocal functions" are:

csc=hypotenuseoppositesec=hypotenuseadjacentcot=adjacentopposite

Example 3.1.3.2

State the reciprocal function of cosecant.

Solution

The reciprocal function of cosecant is sine.

Example 3.1.3.3

Find the value of the expression using a reciprocal identity.

secθ=2π, cosθ=?

Solution

These functions are reciprocals, and the reciprocal of 2π is π2.

Example 3.1.3.4

Find the value of the expression using a reciprocal identity.

cscθ=4, cosθ=?

Solution

These functions are reciprocals, and the reciprocal of 4 is 14.

Review

  1. State the reciprocal function of secant.
  2. State the reciprocal function of cotangent.
  3. State the reciprocal function of sine.

Find the value of the expression using a reciprocal identity.

  1. sinθ=12, cscθ=?
  2. cosθ=32, secθ=?
  3. tanθ=1, cotθ=?
  4. secθ=2, cosθ=?
  5. cscθ=2, sinθ=?
  6. cotθ=1, tanθ=?
  7. sinθ=32, cscθ=?
  8. cosθ=0, secθ=?
  9. tanθ=undefined, cotθ=?
  10. cscθ=233, sinθ=?
  11. sinθ=12 and tanθ=33,cosθ=?
  12. cosθ=22 and tanθ=1, sinθ=?

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.21.

Vocabulary

Term Definition
domain The domain of a function is the set of x-values for which the function is defined.
Range The range of a function is the set of y values for which the function is defined.
Reciprocal Trig Function A reciprocal trigonometric function is a function that is the reciprocal of a typical trigonometric function. For example, since sinx=oppositehypotenuse, the reciprocal function is cscx=hypotenuseopposite

Additional Resources

Interactive Element

Video: The Reciprocal, Quotient, and Pythagorean Identities


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