The Central Idea
The three reciprocal trigonometric functions are defined by reversing the ratios for sine, cosine, and tangent:
\[ \boxed{ \csc\theta=\frac{1}{\sin\theta}, \qquad \sec\theta=\frac{1}{\cos\theta}, \qquad \cot\theta=\frac{1}{\tan\theta} }. \]The corresponding reciprocal pairs are:
- sine and cosecant;
- cosine and secant;
- tangent and cotangent.
Each pair multiplies to \(1\) whenever both functions are defined.
What Is a Reciprocal?
The reciprocal of a nonzero number \(a\) is
\[ \frac{1}{a}. \]For example, the reciprocal of
\[ \frac{3}{5} \]is
\[ \frac{5}{3}. \]Their product is
\[ \frac35\cdot\frac53=1. \]Reciprocal trigonometric functions use exactly this arithmetic idea.
Cosecant
Cosecant, abbreviated \(\csc\), is the reciprocal of sine:
\[ \boxed{\csc\theta=\frac{1}{\sin\theta}}. \]Since
\[ \sin\theta=\frac{O}{H}, \]reversing the ratio gives
\[ \boxed{\csc\theta=\frac{H}{O}}. \]Secant
Secant, abbreviated \(\sec\), is the reciprocal of cosine:
\[ \boxed{\sec\theta=\frac{1}{\cos\theta}}. \]Since
\[ \cos\theta=\frac{A}{H}, \]reversing the ratio gives
\[ \boxed{\sec\theta=\frac{H}{A}}. \]Cotangent
Cotangent, abbreviated \(\cot\), is the reciprocal of tangent:
\[ \boxed{\cot\theta=\frac{1}{\tan\theta}}. \]Since
\[ \tan\theta=\frac{O}{A}, \]reversing the ratio gives
\[ \boxed{\cot\theta=\frac{A}{O}}. \]The Six Right-Triangle Ratios
For an acute angle \(\theta\), all six trigonometric functions can be written in terms of the same three side roles:
\[ \boxed{ \begin{aligned} \sin\theta&=\frac{O}{H}, &\qquad \csc\theta&=\frac{H}{O},\\[4pt] \cos\theta&=\frac{A}{H}, & \sec\theta&=\frac{H}{A},\\[4pt] \tan\theta&=\frac{O}{A}, & \cot\theta&=\frac{A}{O}. \end{aligned} } \]The second function in each row is obtained by reversing the first ratio.
A \(3\)-\(4\)-\(5\) Example
Consider a right triangle with side lengths
\[ 3,\qquad4,\qquad5. \]Let \(\theta\) be opposite the side of length \(3\). Then
\[ \sin\theta=\frac35, \qquad \cos\theta=\frac45, \qquad \tan\theta=\frac34. \]Taking reciprocals gives
\[ \boxed{\csc\theta=\frac53}, \] \[ \boxed{\sec\theta=\frac54}, \]and
\[ \boxed{\cot\theta=\frac43}. \]Reciprocal Products
Reciprocal pairs multiply to \(1\):
\[ \boxed{\sin\theta\csc\theta=1}, \] \[ \boxed{\cos\theta\sec\theta=1}, \]and
\[ \boxed{\tan\theta\cot\theta=1}, \]whenever the functions involved are defined.
Recovering the Original Functions
Reciprocal relationships work in both directions.
From
\[ \csc\theta=\frac{1}{\sin\theta}, \]we also have
\[ \boxed{\sin\theta=\frac{1}{\csc\theta}}. \]Likewise,
\[ \boxed{\cos\theta=\frac{1}{\sec\theta}} \]and
\[ \boxed{\tan\theta=\frac{1}{\cot\theta}}. \]Reciprocal Is Not the Same as Inverse
One of the most important notation distinctions in trigonometry is the difference between a reciprocal and an inverse function.
The reciprocal of sine is
\[ \frac{1}{\sin\theta}=\csc\theta. \]The inverse sine function is written
\[ \sin^{-1}(x) \]or
\[ \arcsin(x). \]These are not the same operation.
Why the Notation Can Be Confusing
In ordinary algebra,
\[ x^{-1}=\frac1x. \]But in trigonometric function notation,
\[ \sin^{-1}(x) \]conventionally means inverse sine, not reciprocal sine.
To write the reciprocal of sine clearly, use
\[ \csc x \]or
\[ \frac{1}{\sin x}. \]From Tangent to Cotangent
Since
\[ \tan\theta=\frac{\sin\theta}{\cos\theta}, \]taking the reciprocal gives
\[ \boxed{ \cot\theta=\frac{\cos\theta}{\sin\theta} }. \]Thus cotangent can be understood both as
\[ \frac{A}{O} \]and as
\[ \frac{\cos\theta}{\sin\theta}. \]The Quotient Relationships
Tangent and cotangent can both be written using sine and cosine:
\[ \boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}} \]and
\[ \boxed{\cot\theta=\frac{\cos\theta}{\sin\theta}}. \]These are called quotient identities.
Finding All Six Functions from a Triangle
Suppose a right triangle has
\[ O=5, \qquad A=12, \qquad H=13. \]Then
\[ \sin\theta=\frac5{13}, \qquad \cos\theta=\frac{12}{13}, \qquad \tan\theta=\frac5{12}. \]Their reciprocals are
\[ \boxed{ \csc\theta=\frac{13}{5}, \qquad \sec\theta=\frac{13}{12}, \qquad \cot\theta=\frac{12}{5} }. \]Finding All Six Functions from One Ratio
Suppose \(\theta\) is acute and
\[ \sin\theta=\frac8{17}. \]Interpret this as
\[ O=8, \qquad H=17. \]Use the Pythagorean relationship:
\[ A=\sqrt{17^2-8^2} = \sqrt{225} = 15. \]Therefore,
\[ \sin\theta=\frac8{17}, \qquad \cos\theta=\frac{15}{17}, \qquad \tan\theta=\frac8{15}, \]and
\[ \boxed{ \csc\theta=\frac{17}{8}, \qquad \sec\theta=\frac{17}{15}, \qquad \cot\theta=\frac{15}{8} }. \]Finding All Six Functions from Cosine
Suppose \(\theta\) is acute and
\[ \cos\theta=\frac7{25}. \]Then
\[ A=7, \qquad H=25. \]The opposite side is
\[ O=\sqrt{25^2-7^2} = \sqrt{576} = 24. \]Thus,
\[ \sin\theta=\frac{24}{25}, \qquad \cos\theta=\frac7{25}, \qquad \tan\theta=\frac{24}{7}, \]and
\[ \boxed{ \csc\theta=\frac{25}{24}, \qquad \sec\theta=\frac{25}{7}, \qquad \cot\theta=\frac7{24} }. \]Finding All Six Functions from Tangent
Suppose \(\theta\) is acute and
\[ \tan\theta=\frac34. \]Let
\[ O=3, \qquad A=4. \]Then
\[ H=\sqrt{3^2+4^2}=5. \]Therefore,
\[ \boxed{ \sin\theta=\frac35,\quad \cos\theta=\frac45,\quad \tan\theta=\frac34,\quad \csc\theta=\frac53,\quad \sec\theta=\frac54,\quad \cot\theta=\frac43 }. \]Special-Angle Reciprocal Values
Reciprocal functions inherit exact values from sine, cosine, and tangent.
\[ \begin{array}{c|ccc} \theta & \csc\theta & \sec\theta & \cot\theta\\ \hline 30^\circ & 2 & \frac{2\sqrt3}{3} & \sqrt3\\[4pt] 45^\circ & \sqrt2 & \sqrt2 & 1\\[4pt] 60^\circ & \frac{2\sqrt3}{3} & 2 & \frac{\sqrt3}{3} \end{array} \]These values can be obtained simply by taking reciprocals of the familiar special-angle values.
Example: Cosecant of \(30^\circ\)
Since
\[ \sin30^\circ=\frac12, \]its reciprocal is
\[ \boxed{\csc30^\circ=2}. \]Example: Secant of \(60^\circ\)
Since
\[ \cos60^\circ=\frac12, \]we have
\[ \boxed{\sec60^\circ=2}. \]Example: Cotangent of \(45^\circ\)
Since
\[ \tan45^\circ=1, \]its reciprocal is also \(1\):
\[ \boxed{\cot45^\circ=1}. \]Rationalizing Reciprocal Values
Suppose
\[ \cos30^\circ=\frac{\sqrt3}{2}. \]Then
\[ \sec30^\circ = \frac{2}{\sqrt3}. \]Rationalizing the denominator gives
\[ \frac{2}{\sqrt3} \cdot \frac{\sqrt3}{\sqrt3} = \boxed{\frac{2\sqrt3}{3}}. \]Both forms are mathematically equivalent.
Magnitude of Cosecant for Acute Angles
For an acute angle,
\[ 0<\sin\theta<1. \]Taking reciprocals therefore gives
\[ \boxed{\csc\theta>1}. \]This is a useful reasonableness check in right-triangle problems.
Magnitude of Secant for Acute Angles
Since
\[ 0<\cos\theta<1 \]for acute \(\theta\),
\[ \boxed{\sec\theta>1}. \]A positive secant value below \(1\) cannot correspond to an acute right-triangle angle.
Cotangent Can Be Smaller or Larger than \(1\)
Since
\[ \cot\theta=\frac{A}{O}, \]cotangent depends on which leg is longer.
For
\[ 0^\circ<\theta<45^\circ, \]the adjacent side is longer, so
\[ \cot\theta>1. \]At \(45^\circ\),
\[ \cot45^\circ=1. \]For
\[ 45^\circ<\theta<90^\circ, \]we have
\[ \cot\theta<1. \]What Happens When a Denominator Is Zero?
A reciprocal cannot be formed when the original quantity is zero.
For example,
\[ \frac10 \]is undefined.
Therefore,
\[ \csc\theta \]is undefined whenever
\[ \sin\theta=0. \]Similar restrictions apply to secant and cotangent.
When Cosecant Is Undefined
Since
\[ \csc\theta=\frac{1}{\sin\theta}, \]cosecant is undefined when
\[ \boxed{\sin\theta=0}. \]In the strictly acute right-triangle setting, sine is never zero, so cosecant is always defined there.
The unit-circle framework will later show exactly where sine becomes zero for general angles.
When Secant Is Undefined
Since
\[ \sec\theta=\frac{1}{\cos\theta}, \]secant is undefined when
\[ \boxed{\cos\theta=0}. \]Again, cosine is positive and nonzero for strictly acute right-triangle angles, but it can be zero once the angle domain is expanded.
When Cotangent Is Undefined
Since
\[ \cot\theta=\frac{\cos\theta}{\sin\theta}, \]cotangent is undefined when
\[ \boxed{\sin\theta=0}. \]Equivalently, the triangle ratio
\[ \cot\theta=\frac{A}{O} \]would require division by zero if the opposite quantity were zero.
Tangent and Secant Share a Restriction
Tangent satisfies
\[ \tan\theta=\frac{\sin\theta}{\cos\theta}, \]while secant satisfies
\[ \sec\theta=\frac1{\cos\theta}. \]Both therefore become undefined whenever
\[ \boxed{\cos\theta=0}. \]This shared denominator will later explain why tangent and secant have related vertical asymptotes.
Cotangent and Cosecant Share a Restriction
Cotangent satisfies
\[ \cot\theta=\frac{\cos\theta}{\sin\theta}, \]while cosecant satisfies
\[ \csc\theta=\frac1{\sin\theta}. \]Both are therefore undefined whenever
\[ \boxed{\sin\theta=0}. \]Reciprocals and Large Values
If a positive number is close to zero, its reciprocal is large.
For example,
\[ \frac{1}{0.1}=10 \]and
\[ \frac{1}{0.01}=100. \]Therefore, whenever sine or cosine approaches zero, its reciprocal function grows very large in magnitude.
This reciprocal behavior becomes visually important in trigonometric graphs.
Reciprocal Relationships and the Pythagorean Identity
Recall
\[ \sin^2\theta+\cos^2\theta=1. \]Dividing this identity by \(\cos^2\theta\), where \(\cos\theta\neq0\), gives
\[ \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}. \]Therefore,
\[ \boxed{\tan^2\theta+1=\sec^2\theta}. \]Another Pythagorean Relationship
Divide
\[ \sin^2\theta+\cos^2\theta=1 \]by \(\sin^2\theta\), where \(\sin\theta\neq0\):
\[ 1+\frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}. \]Therefore,
\[ \boxed{1+\cot^2\theta=\csc^2\theta}. \]The Three Pythagorean Forms
The basic Pythagorean trigonometric identity has three closely related forms:
\[ \boxed{\sin^2\theta+\cos^2\theta=1}, \] \[ \boxed{1+\tan^2\theta=\sec^2\theta}, \]and
\[ \boxed{1+\cot^2\theta=\csc^2\theta}. \]Later lessons on trigonometric identities will use these relationships extensively.
Example: Find Secant from Tangent
Suppose \(\theta\) is acute and
\[ \tan\theta=\frac34. \]Then
\[ \tan^2\theta+1=\sec^2\theta \]gives
\[ \left(\frac34\right)^2+1 = \sec^2\theta. \] \[ \frac9{16}+\frac{16}{16} = \frac{25}{16}. \]Since \(\theta\) is acute, secant is positive:
\[ \boxed{\sec\theta=\frac54}. \]Example: Find Cosecant from Cotangent
Suppose \(\theta\) is acute and
\[ \cot\theta=\frac{12}{5}. \]Use
\[ 1+\cot^2\theta=\csc^2\theta. \]Then
\[ 1+\left(\frac{12}{5}\right)^2 = \csc^2\theta. \] \[ \frac{25}{25}+\frac{144}{25} = \frac{169}{25}. \]Thus,
\[ \boxed{\csc\theta=\frac{13}{5}}. \]Complementary Relationships
Sine and cosine exchange roles for complementary angles:
\[ \sin\theta=\cos(90^\circ-\theta). \]Taking reciprocals gives
\[ \boxed{ \csc\theta=\sec(90^\circ-\theta) }. \]Similarly,
\[ \boxed{ \sec\theta=\csc(90^\circ-\theta) }. \]Tangent and Cotangent as Cofunctions
For complementary acute angles,
\[ \tan\theta=\cot(90^\circ-\theta) \]and
\[ \boxed{ \cot\theta=\tan(90^\circ-\theta) }. \]These cofunction relationships arise because opposite and adjacent sides switch roles between the two acute angles of a right triangle.
Reciprocal Values on a Calculator
Many calculators provide direct buttons for sine, cosine, and tangent but not for cosecant, secant, and cotangent.
In that case, evaluate reciprocally:
\[ \csc\theta=\frac1{\sin\theta}, \] \[ \sec\theta=\frac1{\cos\theta}, \]and
\[ \cot\theta=\frac1{\tan\theta}. \]Example: Calculator Evaluation
To estimate
\[ \sec37^\circ, \]calculate
\[ \frac1{\cos37^\circ}. \]Since
\[ \cos37^\circ\approx0.7986, \]we obtain
\[ \boxed{\sec37^\circ\approx1.252}. \]Preserve Precision When Taking Reciprocals
Suppose a calculator reports
\[ \sin23^\circ\approx0.390731\ldots \]If you need \(\csc23^\circ\), it is better to calculate
\[ \frac1{\sin23^\circ} \]directly rather than first rounding sine to \(0.39\).
Early rounding is amplified when reciprocals are taken.
Applied Interpretation of Secant
Suppose a segment of length \(r\) makes an acute angle \(\theta\) with the horizontal, and its horizontal component is \(x\).
Since
\[ \cos\theta=\frac{x}{r}, \]secant gives
\[ \sec\theta=\frac{r}{x}. \]Secant therefore compares the total directed length with its horizontal component.
Applied Interpretation of Cosecant
If the vertical component is \(y\), then
\[ \sin\theta=\frac{y}{r}. \]Therefore,
\[ \csc\theta=\frac{r}{y}. \]Cosecant compares the total length with its vertical component.
Applied Interpretation of Cotangent
Tangent measures
\[ \frac{\text{rise}}{\text{run}}. \]Cotangent reverses that comparison:
\[ \boxed{ \cot\theta= \frac{\text{run}}{\text{rise}} }. \]In some applications, horizontal distance per unit vertical change is more useful than vertical change per unit horizontal distance.
Reciprocal Functions Are Not Redundant
Because cosecant, secant, and cotangent can be expressed using sine, cosine, and tangent, they do not contain fundamentally new information.
However, they are still useful because they:
- make reciprocal structure visible;
- simplify many identities;
- describe useful ratios directly;
- appear naturally in calculus;
- clarify domain restrictions and asymptotic behavior;
- provide compact notation for repeated reciprocals.
Why Reciprocal Structure Matters Later
Reciprocal relationships become increasingly useful when trigonometric expressions become more complicated.
For example,
\[ \frac1{\cos^2\theta} \]can be written compactly as
\[ \sec^2\theta. \]Likewise,
\[ \frac{\cos\theta}{\sin\theta} \]becomes
\[ \cot\theta. \]Recognizing these forms is essential in later algebraic simplification.
Common Error: Pairing the Wrong Functions
The correct reciprocal pairs are
\[ \boxed{ \sin\leftrightarrow\csc,\qquad \cos\leftrightarrow\sec,\qquad \tan\leftrightarrow\cot }. \]Secant is not the reciprocal of sine, and cosecant is not the reciprocal of cosine.
Common Error: Reversing the Wrong Side Ratio
If
\[ \sin\theta=\frac{O}{H}, \]then
\[ \csc\theta=\frac{H}{O}. \]Writing
\[ \csc\theta=\frac{H}{A} \]mistakenly uses the secant ratio.
Common Error: Confusing Reciprocal and Opposite
A reciprocal does not change the sign of a number.
For example, the reciprocal of
\[ \frac35 \]is
\[ \frac53, \]not
\[ -\frac35. \]Taking an opposite and taking a reciprocal are different operations.
Common Error: Forgetting Undefined Cases
The expression
\[ \frac1{\sin\theta} \]only exists when
\[ \sin\theta\neq0. \]Reciprocal identities always carry the implicit requirement that the denominator be nonzero.
Common Error: Using an Inverse Key for a Reciprocal
On many calculators, a key labeled
\[ \sin^{-1} \]computes inverse sine.
It does not compute cosecant.
To calculate cosecant when no dedicated key is present, use
\[ \frac1{\sin\theta}. \]Common Error: Losing the Square in an Identity
The identity
\[ 1+\tan^2\theta=\sec^2\theta \]does not say
\[ 1+\tan\theta=\sec\theta. \]The squared functions are essential.
Common Error: Taking a Reciprocal after Rounding Too Much
If
\[ \cos\theta\approx0.08, \]then its reciprocal is already relatively large:
\[ \frac1{0.08}=12.5. \]Small rounding errors in a quantity near zero can create much larger changes in its reciprocal.
Preserve calculator precision whenever possible.
A Reliable Procedure for Reciprocal Functions
- Identify the original trigonometric function.
- Recall its reciprocal partner.
- Reverse the numerical ratio.
- Check that the original quantity is nonzero.
- Simplify the resulting fraction.
- Rationalize a denominator if appropriate.
- Check that the result is reasonable.
A Reliable Procedure for Finding All Six Functions
- Translate the given ratio into two side lengths.
- Use the Pythagorean relationship to find the third side if needed.
- Write sine, cosine, and tangent from the side roles.
- Reverse each ratio to obtain cosecant, secant, and cotangent.
- Simplify all fractions.
- Check reciprocal products when useful.
A Reliable Procedure for Using Reciprocal Identities
- Identify which trigonometric functions appear.
- Replace a reciprocal function with \(1\) over its partner when useful.
- Replace tangent or cotangent with sine-and-cosine quotients when useful.
- Track denominator restrictions.
- Simplify algebraically.
- Verify with an equivalent form or numerical example if needed.
Practice 1: Reciprocal Pairs
Name the reciprocal partner of each function:
\[ \sin\theta,\qquad \cos\theta,\qquad \tan\theta. \]Practice 2: Find the Reciprocals
Suppose
\[ \sin\theta=\frac5{13}, \qquad \cos\theta=\frac{12}{13}, \qquad \tan\theta=\frac5{12}. \]Find \(\csc\theta\), \(\sec\theta\), and \(\cot\theta\).
Practice 3: Find All Six Functions
Suppose \(\theta\) is acute and
\[ \sin\theta=\frac{12}{13}. \]Find all six trigonometric functions.
Practice 4: Start from Tangent
Suppose \(\theta\) is acute and
\[ \tan\theta=\frac8{15}. \]Find \(\sin\theta\), \(\cos\theta\), \(\csc\theta\), \(\sec\theta\), and \(\cot\theta\).
Practice 5: Exact Special-Angle Values
Evaluate exactly:
\[ \csc30^\circ, \qquad \sec60^\circ, \qquad \cot45^\circ. \]Practice 6: Quotient Identity
Suppose
\[ \sin\theta=\frac35 \qquad\text{and}\qquad \cos\theta=\frac45. \]Use the quotient identity to find \(\cot\theta\).
Practice 7: Pythagorean Identity
Suppose \(\theta\) is acute and
\[ \tan\theta=\frac5{12}. \]Use
\[ 1+\tan^2\theta=\sec^2\theta \]to find \(\sec\theta\).
Practice 8: Another Pythagorean Identity
Suppose \(\theta\) is acute and
\[ \cot\theta=\frac34. \]Find \(\csc\theta\).
Practice 9: Undefined Values
If
\[ \cos\theta=0, \]which of the following are necessarily undefined?
\[ \sec\theta,\qquad \tan\theta,\qquad \csc\theta. \]Practice 10: Identify the Error
A student writes
\[ \sin^{-1}\theta=\csc\theta. \]Explain why this notation is incorrect.
Practice Answers
Practice 1
\[ \boxed{ \sin\theta\leftrightarrow\csc\theta, \qquad \cos\theta\leftrightarrow\sec\theta, \qquad \tan\theta\leftrightarrow\cot\theta }. \]Practice 2
\[ \boxed{ \csc\theta=\frac{13}{5}, \qquad \sec\theta=\frac{13}{12}, \qquad \cot\theta=\frac{12}{5} }. \]Practice 3
If
\[ O=12,\qquad H=13, \]then
\[ A=\sqrt{13^2-12^2}=5. \]Therefore,
\[ \boxed{ \sin\theta=\frac{12}{13},\quad \cos\theta=\frac5{13},\quad \tan\theta=\frac{12}{5},\quad \csc\theta=\frac{13}{12},\quad \sec\theta=\frac{13}{5},\quad \cot\theta=\frac5{12} }. \]Practice 4
Let
\[ O=8,\qquad A=15. \]Then
\[ H=\sqrt{8^2+15^2}=17. \]Thus,
\[ \boxed{ \sin\theta=\frac8{17},\quad \cos\theta=\frac{15}{17},\quad \csc\theta=\frac{17}{8},\quad \sec\theta=\frac{17}{15},\quad \cot\theta=\frac{15}{8} }. \]Practice 5
\[ \boxed{\csc30^\circ=2} \] \[ \boxed{\sec60^\circ=2} \] \[ \boxed{\cot45^\circ=1}. \]Practice 6
\[ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{4/5}{3/5} = \boxed{\frac43}. \]Practice 7
\[ 1+\left(\frac5{12}\right)^2=\sec^2\theta. \] \[ 1+\frac{25}{144} = \frac{169}{144}. \]Since \(\theta\) is acute,
\[ \boxed{\sec\theta=\frac{13}{12}}. \]Practice 8
\[ 1+\left(\frac34\right)^2=\csc^2\theta. \] \[ 1+\frac9{16} = \frac{25}{16}. \]Thus,
\[ \boxed{\csc\theta=\frac54}. \]Practice 9
If
\[ \cos\theta=0, \]then both
\[ \sec\theta=\frac1{\cos\theta} \]and
\[ \tan\theta=\frac{\sin\theta}{\cos\theta} \]are undefined.
Therefore,
\[ \boxed{\sec\theta\text{ and }\tan\theta\text{ are necessarily undefined}.} \]Cosecant depends on \(\sin\theta\), not directly on \(\cos\theta\).
Practice 10
The notation
\[ \sin^{-1}(x) \]denotes inverse sine, which returns an angle from a sine value.
The reciprocal of sine is cosecant:
\[ \boxed{\csc\theta=\frac1{\sin\theta}}. \]Inverse and reciprocal are different operations.
What to Remember
The reciprocal pairs are
\[ \boxed{ \sin\theta\leftrightarrow\csc\theta, \qquad \cos\theta\leftrightarrow\sec\theta, \qquad \tan\theta\leftrightarrow\cot\theta }. \]Their defining relationships are
\[ \boxed{ \csc\theta=\frac1{\sin\theta}, \qquad \sec\theta=\frac1{\cos\theta}, \qquad \cot\theta=\frac1{\tan\theta} }. \]- Cosecant is hypotenuse over opposite.
- Secant is hypotenuse over adjacent.
- Cotangent is adjacent over opposite.
- Reciprocal pairs multiply to \(1\) when defined.
- Cotangent also satisfies \(\cot\theta=\cos\theta/\sin\theta\).
- A reciprocal function is undefined when its denominator is zero.
- Reciprocal and inverse functions are not the same thing.
- The reciprocal functions generate two additional Pythagorean identities.
Where We Go Next
Right-triangle ratios introduced all six trigonometric functions, but the triangle model by itself is limited to acute angles.
The next lesson, The Unit Circle, places trigonometry in the coordinate plane. This makes it possible to define sine, cosine, tangent, and their reciprocals for angles far beyond the acute right-triangle setting.