1.21: Reciprocal Identities
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?
At the end of this Concept, you'll be able to list these ratios, as well as know what they are called.
Watch This
The first portion of this video will help you understand reciprocal functions.
James Sousa: The Reciprocal, Quotient, and Pythagorean Identities
Guidance
A reciprocal of a fraction
First, consider the definition of the sine function for angles of rotation:
Analogously, the cosine function and the secant function are reciprocals, and the tangent and cotangent function are reciprocals:
Example A
Find the value of the expression using a reciprocal identity.
Solution:
These functions are reciprocals, so if
Example B
Find the value of the expression using a reciprocal identity.
Solution: These functions are reciprocals, and the reciprocal of
We can also use the reciprocal relationships to determine the domain and range of functions.
Example C
Find the value of the expression using a reciprocal identity.
Solution: These functions are reciprocals, and the reciprocal of
Vocabulary
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 trig function is a relationship that is the reciprocal of a typical trig function. For example, since
Guided Practice
1. State the reciprocal function of cosecant.
2. Find the value of the expression using a reciprocal identity.
3. Find the value of the expression using a reciprocal identity.
Solutions:
1. The reciprocal function of cosecant is sine.
2. These functions are reciprocals, and the reciprocal of
3. These functions are reciprocals, and the reciprocal of
Concept Problem Solution
Since the three regular trig functions are defined as:
then the three functions  called "reciprocal functions" are:
Practice
 State the reciprocal function of secant.
 State the reciprocal function of cotangent.
 State the reciprocal function of sine.
Find the value of the expression using a reciprocal identity.

sinθ=12,cscθ=? 
cosθ=−3√2,secθ=? 
tanθ=1,cotθ=? 
secθ=2√,cosθ=? 
cscθ=2,sinθ=? 
cotθ=−1,tanθ=? 
sinθ=3√2,cscθ=? 
cosθ=0,secθ=? 
tanθ= undefined,cotθ=? 
cscθ=23√3,sinθ=? 
sinθ=−12 andtanθ=3√3,cosθ=? 
cosθ=2√2 andtanθ=1,sinθ=?
Notes/Highlights Having trouble? Report an issue.
Color  Highlighted Text  Notes  

Show More 
domain
The domain of a function is the set of values for which the function is defined.Range
The range of a function is the set of 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 , the reciprocal function isImage Attributions
Here you'll learn what the reciprocal trig functions are, and how they relate to the sine, cosine, and tangent functions.