# 1.21: Reciprocal Identities

Difficulty Level: At Grade Created by: CK-12
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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.

### Guidance

A reciprocal of a fraction \begin{align*}\frac{a}{b}\end{align*} is the fraction \begin{align*}\frac{b}{a}\end{align*}. 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: \begin{align*}\sin \theta = \frac{y}{r}\end{align*}. Now consider the cosecant function: \begin{align*}\csc \theta = \frac{r}{y}\end{align*}. In the unit circle, these values are \begin{align*}\sin \theta = \frac{y}{1} = y\end{align*} and \begin{align*}\csc \theta = \frac{1}{y}\end{align*}. 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 \begin{align*}\sin \theta = \frac{1}{2}\end{align*}, then \begin{align*}\csc \theta = \frac{2}{1} = 2\end{align*}.

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

\begin{align*}\sec \theta = \frac{1}{\cos \theta} && \text{or} && \cos \theta = \frac{1}{\sec \theta}\\ \cot \theta = \frac{1}{\tan \theta} && \text{or} && \tan \theta = \frac{1}{\cot \theta}\end{align*}

#### Example A

Find the value of the expression using a reciprocal identity.

\begin{align*}\cos \theta = .3, \sec \theta = ?\end{align*}

Solution: \begin{align*}\sec \theta = \frac{10}{3}\end{align*}

These functions are reciprocals, so if \begin{align*}\cos \theta = .3\end{align*}, then \begin{align*}\sec \theta = \frac{1}{.3}\end{align*}. It is easier to find the reciprocal if we express the values as fractions: \begin{align*}\cos \theta = .3 = \frac{3}{10} \Rightarrow \sec \theta = \frac{10}{3}\end{align*}.

#### Example B

Find the value of the expression using a reciprocal identity.

\begin{align*}\cot \theta = \frac{4}{3}, \tan \theta = ?\end{align*}

Solution: These functions are reciprocals, and the reciprocal of \begin{align*}\frac{4}{3}\end{align*} is \begin{align*}\frac{3}{4}\end{align*}.

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.

\begin{align*}\sin \theta = \frac{1}{2}, \csc \theta = ?\end{align*}

Solution: These functions are reciprocals, and the reciprocal of \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}2\end{align*}.

### 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 \begin{align*}\sin x = \frac{opposite}{hypotenuse}\end{align*}, the reciprocal function is \begin{align*}\csc x = \frac{hypotenuse}{opposite}\end{align*}

### Guided Practice

1. State the reciprocal function of cosecant.

2. Find the value of the expression using a reciprocal identity.

\begin{align*}\sec \theta = \frac{2}{\pi}, \cos \theta = ?\end{align*}

3. Find the value of the expression using a reciprocal identity.

\begin{align*}\csc \theta = 4, \cos \theta = ?\end{align*}

Solutions:

1. The reciprocal function of cosecant is sine.

2. These functions are reciprocals, and the reciprocal of \begin{align*}\frac{2}{\pi}\end{align*} is \begin{align*}\frac{\pi}{2}\end{align*}.

3. These functions are reciprocals, and the reciprocal of \begin{align*}4\end{align*} is \begin{align*}\frac{1}{4}\end{align*}.

### Concept Problem Solution

Since the three regular trig functions are defined as:

\begin{align*} \sin = \frac{opposite}{hypotenuse}\\ \cos = \frac{adjacent}{hypotenuse}\\ \tan = \frac{opposite}{adjacent}\\ \end{align*}

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

\begin{align*} \csc = \frac{hypotenuse}{opposite}\\ \sec = \frac{hypotenuse}{adjacent}\\ \cot = \frac{adjacent}{opposite}\\ \end{align*}

### Practice

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. \begin{align*}\sin \theta= \frac{1}{2}, \csc \theta =?\end{align*}
2. \begin{align*}\cos \theta= -\frac{\sqrt{3}}{2}, \sec \theta =?\end{align*}
3. \begin{align*}\tan \theta= 1, \cot \theta =?\end{align*}
4. \begin{align*}\sec \theta= \sqrt{2}, \cos \theta =?\end{align*}
5. \begin{align*}\csc \theta= 2, \sin \theta =?\end{align*}
6. \begin{align*}\cot \theta= -1, \tan \theta =?\end{align*}
7. \begin{align*}\sin \theta= \frac{\sqrt{3}}{2}, \csc \theta =?\end{align*}
8. \begin{align*}\cos \theta= 0, \sec \theta =?\end{align*}
9. \begin{align*}\tan \theta=\end{align*}undefined\begin{align*}, \cot \theta =?\end{align*}
10. \begin{align*}\csc \theta= \frac{2\sqrt{3}}{3}, \sin \theta =?\end{align*}
11. \begin{align*}\sin \theta= \frac{-1}{2}\end{align*} and \begin{align*}\tan \theta= \frac{\sqrt{3}}{3}, \cos \theta=?\end{align*}
12. \begin{align*}\cos \theta= \frac{\sqrt{2}}{2}\end{align*} and \begin{align*}\tan \theta= 1, \sin \theta=?\end{align*}

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### Vocabulary Language: English

TermDefinition
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 $\sin x = \frac{opposite}{hypotenuse}$, the reciprocal function is $\csc x = \frac{hypotenuse}{opposite}$

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