Every function has its mirror image. For sine, cosine, and tangent, those mirrors are cosecant, secant, and cotangent. These reciprocal functions appear in physics, engineering, and astronomy whenever quantities are inversely related. In this lesson you will learn their definitions, how to evaluate them, and where they are undefined.
Today's hook, If $\sin \theta = 0$, then $\frac{1}{\sin \theta}$ is undefined because you cannot divide by zero. Where on the unit circle does $\sin \theta = 0$? And what does this tell you about where $\csc \theta$ is undefined?
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If $\sin \theta = 0$, then $\frac{1}{\sin \theta}$ is undefined because you cannot divide by zero. Where on the unit circle does $\sin \theta = 0$? And what does this tell you about where $\csc \theta$ is undefined?
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The three reciprocal functions
Work through the core explanation before applying it.
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The three reciprocal functions
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For every basic trigonometric function there is a corresponding reciprocal function. A number and its reciprocal always have the same sign, so ASTC applies directly to the reciprocal functions too.
The reciprocal functions are defined by flipping the original function. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. They are undefined wherever their denominator equals zero.
csc · sec · cot
Cosecant
$\csc \theta = \frac{1}{\sin \theta}$. Undefined when $\sin \theta = 0$, i.e. $\theta = n\pi$.
Secant
$\sec \theta = \frac{1}{\cos \theta}$. Undefined when $\cos \theta = 0$, i.e. $\theta = \frac{\pi}{2} + n\pi$.
Cosecant$\csc \theta = \frac{1}{\sin \theta}$, the reciprocal of sine.
Secant$\sec \theta = \frac{1}{\cos \theta}$, the reciprocal of cosine.
Cotangent$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$, the reciprocal of tangent.
Reciprocal identityA relationship connecting a trig function with its reciprocal.
DomainThe set of all input values for which a function is defined.
RangeThe set of all possible output values of a function.
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Domains and signs of reciprocal functions
Work through the core explanation before applying it.
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Domains and signs of reciprocal functions
Because division by zero is undefined, each reciprocal function has restrictions on its domain. The signs follow directly from the original function via ASTC:
$\csc \theta$ is positive where $\sin \theta$ is positive (Quadrants I and II)
$\sec \theta$ is positive where $\cos \theta$ is positive (Quadrants I and IV)
$\cot \theta$ is positive where $\tan \theta$ is positive (Quadrants I and III)
The range of cosecant and secant is $(-\infty, -1] \cup [1, \infty)$ because they are reciprocals of numbers in $[-1, 1]$ (excluding 0). The range of cotangent is all real numbers.
Why astronomers use parallax and secants. When measuring the distance to nearby stars, astronomers use the parallax method. The small angle of shift is related to the star's distance through a reciprocal relationship, essentially a secant function. Without understanding reciprocal trig functions, the mathematics of stellar distances would be impossible.
Three reciprocal definitions: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$; Signs follow ASTC, a reciprocal has the same sign as its original function
Pause, copy the three reciprocal definitions ($\csc\theta = 1/\sin\theta$; $\sec\theta = 1/\cos\theta$; $\cot\theta = \cos\theta/\sin\theta$) and the sign rule (same sign as the original function) into your book.
Which quadrants have negative $\sec \theta$?
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Worked Example, Evaluating a reciprocal function
+5 XP for trying first
We just saw that $\sec\theta = 1/\cos\theta$ and the reciprocals share the ASTC sign of their originals. That raises a question: how do we actually compute a reciprocal value like $\sec\frac{\pi}{3}$ in practice? This card answers it → evaluate the original function first, then take the reciprocal: $\sec\frac{\pi}{3} = 1/\cos\frac{\pi}{3} = 1/(1/2) = 2$.
Find the exact value of $\sec \frac{\pi}{3}$.
Your turn first. Try it yourself before viewing the solution.
To evaluate a reciprocal trig function: write it as $\frac{1}{\text{original}}$, then substitute the exact value; $\sec \frac{\pi}{3} = \frac{1}{\cos \frac{\pi}{3}} = \frac{1}{1/2} = 2$
Pause, copy the reciprocal evaluation method: write as $1/\text{original}$, substitute exact value, simplify, and the worked result $\sec\frac{\pi}{3} = 2$ into your book.
True or false: $\csc \frac{\pi}{6} = 2$.
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Worked Example, Finding where a reciprocal is undefined
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We just saw that reciprocals are $1/\text{original}$. That raises a question: since division by zero is undefined, at which angles does each reciprocal function fail to exist? This card answers it → $\csc\theta$ is undefined when $\sin\theta = 0$; $\sec\theta$ when $\cos\theta = 0$; $\cot\theta$ when $\sin\theta = 0$.
State the values of $\theta$ in $[0, 2\pi]$ for which $\cot \theta$ is undefined.
Your turn first. Try it yourself before viewing the solution.
SOLUTION
2 marks · HSC band 4+
Step 1: Identify the condition.
$$\cot \theta = \frac{\cos \theta}{\sin \theta} \text{ is undefined when } \sin \theta = 0$$
$\cot \theta$ is undefined when $\sin \theta = 0$, i.e. $\theta = 0, \pi, 2\pi, \ldots$; $\sec \theta$ is undefined when $\cos \theta = 0$, i.e. $\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$
Pause, copy the undefined-angle rules: $\cot\theta$ undefined at $\theta = 0, \pi, 2\pi,\ldots$ (where $\sin\theta = 0$); $\sec\theta$ undefined at $\theta = \frac{\pi}{2}, \frac{3\pi}{2},\ldots$ (where $\cos\theta = 0$) into your book.
Complete the sentence: $\sec \theta$ is undefined when $\cos \theta = $ , which occurs at $\theta = \frac{\pi}{2} + n\pi$.
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Worked Example, Simplifying with reciprocals
+5 XP for trying first
We just saw the domain restrictions of reciprocal functions. That raises a question: when a reciprocal appears inside a larger expression, how do we simplify, do we just substitute the definition? This card answers it → yes: replace with $1/\text{original}$, then cancel pairs like $\sin\theta \cdot \csc\theta = 1$.
Key identity: $\sin \theta \cdot \csc \theta = 1$ and $\cos \theta \cdot \sec \theta = 1$ (where defined); When simplifying, replace reciprocals with $\frac{1}{\text{original}}$ and cancel
Pause, copy the two cancellation identities: $\sin\theta \cdot \csc\theta = 1$ and $\cos\theta \cdot \sec\theta = 1$, plus the simplification strategy (replace with $1/\text{original}$, then cancel) into your book.
Which of the following is the odd one out it does NOT simplify to 1?
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Common traps
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Common traps
Trap 1, Confusing secant and cosecant
Students often swap $\sec \theta$ and $\csc \theta$. Remember the pairings: secant goes with cosine, cosecant goes with sine. The "co-" in cosecant matches sine (no "co-"), while secant matches cosine (which has "co-").
Trap 2, Forgetting that cotangent can be written as $\frac{\cos}{\sin}$
While $\cot \theta = \frac{1}{\tan \theta}$ is correct, it is often more useful to write $\cot \theta = \frac{\cos \theta}{\sin \theta}$, especially when simplifying identities or finding where it is undefined.
Trap 3, Including undefined points in the domain
When asked for the domain of $\sec \theta$, some students say "all real numbers" or only exclude $\frac{\pi}{2}$. They forget that every odd multiple of $\frac{\pi}{2}$ must be excluded. Use the general form $n \in \mathbb{Z}$.
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Drill, build fluency
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Drill, build fluency
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Work these through step-by-step. Use exact values and state any domain restrictions.
Match each function to its correct undefined condition.
$\csc \theta$
$\sec \theta$
$\cot \theta$
undefined when $\sin \theta = 0$
undefined when $\cos \theta = 0$
undefined when $\sin \theta = 0$
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Revisit, the unit circle
Run the quick drill and copy the summary into your book.
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Revisit, the unit circle
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Return to your original answer from Section 01. $\sin \theta = 0$ at the points where the unit circle crosses the $x$-axis: $\theta = 0, \pi, 2\pi$, and so on. At these angles, the $y$-coordinate is 0. Since $\csc \theta = \frac{1}{\sin \theta}$, dividing by zero means $\csc \theta$ is undefined at every integer multiple of $\pi$. These points become vertical asymptotes on the graph of $y = \csc \theta$.
How does your initial thinking compare? Did you identify all three points in $[0, 2\pi]$?
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Multiple choice
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Practice
Multiple choice
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Short answer
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Structured response questions
ApplyBand 4–5
Exact values with ASTC
(a) Find the exact value of $\csc \frac{5\pi}{6}$. (b) Find the exact value of $\sec \frac{7\pi}{4}$. Show reference angles and ASTC reasoning.
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(a) $\frac{5\pi}{6}$ is in QII. Reference angle = $\frac{\pi}{6}$. $\sin \frac{\pi}{6} = \frac{1}{2}$. In QII, sine is positive, so $\sin \frac{5\pi}{6} = \frac{1}{2}$. Therefore $\csc \frac{5\pi}{6} = \mathbf{2}$.
(b) $\frac{7\pi}{4}$ is in QIV. Reference angle = $\frac{\pi}{4}$. $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. In QIV, cosine is positive, so $\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}$. Therefore $\sec \frac{7\pi}{4} = \mathbf{\sqrt{2}}$.
ApplyBand 5–6
Simplify to a single function
Simplify $\frac{\sec \theta}{\tan \theta}$ to a single trigonometric function. State any restrictions on $\theta$.
Restrictions: $\cos \theta \neq 0$ and $\sin \theta \neq 0$, so $\theta \neq \frac{n\pi}{2}$ for any integer $n$.
AnalyseBand 6
Explain the range of cosecant
Explain why the range of $y = \csc \theta$ is $(-\infty, -1] \cup [1, \infty)$. Use the relationship between $\csc \theta$ and $\sin \theta$ in your explanation.
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Since $\csc \theta = \frac{1}{\sin \theta}$ and $\sin \theta \in [-1, 1]$ (excluding 0), the reciprocal of a number in $(-1, 1)$ (excluding 0) lies outside $(-1, 1)$.
When $0 < \sin \theta \leq 1$, we have $\csc \theta \geq 1$.
When $-1 \leq \sin \theta < 0$, we have $\csc \theta \leq -1$.
Therefore the range is $(-\infty, -1] \cup [1, \infty)$.
Key insight: the reciprocal function "flips" the interval. Values close to 0 become very large, while the endpoints $\pm 1$ stay fixed.
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