Reciprocal Trigonometric Functions
The three reciprocal functions — cosecant, secant, and cotangent — are not new trig ratios. They are simply $1/\sin$, $1/\cos$, and $1/\tan$. But they appear constantly in advanced mathematics, proofs, and physics, so fluency with them is essential. In this lesson you'll define them, pin down their domains, evaluate them for exact angles, and understand how to use the Pythagorean identity to find all six trig ratios from a single known value.
If $\sin\theta = \dfrac{3}{5}$, what is $\csc\theta$? What about $\cos\theta$ and $\sec\theta$? Write your gut answer — you may not have enough information yet to be certain of the sign.
There are only two core moves in every reciprocal trig problem. Flip to find the reciprocal, then use the Pythagorean identity to find the missing ratio when only one ratio is given.
Every reciprocal trig question lives on one of two roads: flip the given ratio to get its reciprocal directly, or apply $\sin^2\theta + \cos^2\theta = 1$ to find a missing ratio before flipping.
Key facts
- The definitions $\csc\theta = 1/\sin\theta$, $\sec\theta = 1/\cos\theta$, $\cot\theta = 1/\tan\theta = \cos\theta/\sin\theta$
- The domain restrictions for each reciprocal function
- That reciprocal functions inherit their signs from their parent functions
Concepts
- Why $\csc\theta$ and $\cot\theta$ share the same domain restriction (both require $\sin\theta \neq 0$)
- How the Pythagorean identity links sin and cos to find a missing ratio
- Why we need to know the quadrant to determine the sign of the missing ratio
Skills
- Evaluate exact values of $\csc\theta$, $\sec\theta$, $\cot\theta$ for standard angles
- Find all six trig ratios given one ratio and the quadrant
- Identify where each reciprocal function is undefined in $[0, 2\pi)$
The three reciprocal trigonometric functions are defined as:
Domain restrictions — each function is undefined when its denominator is zero:
- $\csc\theta$ undefined when $\sin\theta = 0$, i.e. $\theta = n\pi$ for integer $n$ ($\theta = 0°, 180°, 360°, \ldots$)
- $\sec\theta$ undefined when $\cos\theta = 0$, i.e. $\theta = \dfrac{\pi}{2} + n\pi$ ($\theta = 90°, 270°, \ldots$)
- $\cot\theta$ undefined when $\sin\theta = 0$, i.e. $\theta = n\pi$ (same as $\csc\theta$)
Sign behaviour: each reciprocal function carries the same ASTC sign as its parent. For example, $\csc\theta$ is positive where $\sin\theta$ is positive (Q1 and Q2), and negative in Q3 and Q4.
= 1/ (undefined at = n); = 1/ (undefined at = /2 + n); = / (undefined at = n); Signs: same sign as (Q1, Q2 positive); same sign as (Q1, Q4 positive); same sign as (Q1, Q3 positive)
Pause — copy all three reciprocal definitions and their domain restrictions into your book: $\csc\theta = 1/\sin\theta$ (undefined at $\theta = n\pi$), $\sec\theta = 1/\cos\theta$ (undefined at $\theta = \pi/2+n\pi$), $\cot\theta = \cos\theta/\sin\theta$ (undefined at $\theta = n\pi$).
Quick check: What is the exact value of $\sec\!\left(\dfrac{\pi}{3}\right)$?
We just saw that $\csc\theta = 1/\sin\theta$, $\sec\theta = 1/\cos\theta$, and $\cot\theta = \cos\theta/\sin\theta$. That raises a question: since each reciprocal function inherits the sign of its parent, do they follow the same ASTC quadrant pattern? This card answers it → $\csc$ is positive wherever $\sin$ is positive, $\sec$ wherever $\cos$ is positive, and $\cot$ wherever $\tan$ is positive — confirmed by the table below.
Because each reciprocal function is just $1 \div (\text{parent function})$, it has exactly the same sign as its parent in every quadrant:
| Function | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| $\sin\theta$ / $\csc\theta$ | + | + | − | − |
| $\cos\theta$ / $\sec\theta$ | + | − | − | + |
| $\tan\theta$ / $\cot\theta$ | + | − | + | − |
Note that $\csc\theta$ and $\cot\theta$ are undefined at the same angles ($\theta = n\pi$), even though $\cot$ is a "tan" reciprocal and $\csc$ is a "sin" reciprocal. Both require $\sin\theta \neq 0$.
Reciprocal functions have the same sign pattern as their parents — no new quadrant analysis required; positive in Q1 and Q2 (where > 0); positive in Q1 and Q4 (where > 0); positive in Q1 and Q3 (where > 0)
Pause — copy the sign table for $\sin/\csc$, $\cos/\sec$, and $\tan/\cot$ across all four quadrants into your book.
Did you get this? True or false: $\csc\theta$ and $\cot\theta$ are undefined at the same values of $\theta$.
Worked examples · 3 in a row, reveal as you go
Given that $\sin\theta = \dfrac{3}{5}$ and $\dfrac{\pi}{2} < \theta < \pi$, find $\csc\theta$ and $\sec\theta$.
$\cos\theta = -\sqrt{1 - \sin^2\theta} = -\sqrt{1 - \tfrac{9}{25}} = -\sqrt{\tfrac{16}{25}} = -\dfrac{4}{5}$
Find the exact value of $\cot\!\left(\dfrac{\pi}{6}\right)$.
For what values of $\theta$ in $[0, 2\pi)$ is $\sec\theta$ undefined? State and explain the reason.
Fill the gap: If $\cos\theta = \dfrac{5}{13}$ and $\sin\theta < 0$, then $\sec\theta = $ .
Misconceptions to fix · the 3 traps that cost marks
Did you get this? True or false: if $\theta$ is in Q4 and $\cos\theta = \dfrac{5}{13}$, then $\sec\theta = \dfrac{13}{5}$ (positive).
Activities · practice with the ideas
If $\cos\theta = \dfrac{5}{13}$ and $\sin\theta < 0$, find $\sec\theta$ and $\csc\theta$.
Find the exact value of $\cot\!\left(\dfrac{\pi}{6}\right)$.
For what values of $\theta$ in $[0, 2\pi)$ is $\sec\theta$ undefined?
Find the exact value of $\csc\!\left(\dfrac{5\pi}{6}\right)$.
Explain why $\csc\theta$ and $\cot\theta$ share the same domain restrictions, even though $\cot$ is a "tangent" reciprocal.
Odd one out: Three of these statements are correct. Which one is NOT?
Earlier you were asked: if $\sin\theta = \dfrac{3}{5}$, what is $\csc\theta$, $\cos\theta$, and $\sec\theta$?
$\csc\theta = \dfrac{5}{3}$ — the direct reciprocal, positive because $\sin\theta > 0$.
For $\cos\theta$: without the quadrant, we cannot determine the sign. If $\theta$ is in Q1, $\cos\theta = +\dfrac{4}{5}$ and $\sec\theta = +\dfrac{5}{4}$. If $\theta$ is in Q2, $\cos\theta = -\dfrac{4}{5}$ and $\sec\theta = -\dfrac{5}{4}$. This is exactly why the quadrant is not optional information — it determines the sign of every missing ratio.
And why do $\csc\theta$ and $\cot\theta$ share domain restrictions? Because $\cot\theta = \cos\theta/\sin\theta$, which has $\sin\theta$ in its denominator — the same denominator as $\csc\theta = 1/\sin\theta$. Both vanish at $\theta = n\pi$.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. If $\cos\theta = \dfrac{5}{13}$ and $\sin\theta < 0$, find $\sec\theta$ and $\csc\theta$. (3 marks)
Q2. Find the exact value of $\cot\!\left(\dfrac{\pi}{6}\right)$. (1 mark)
Q3. For what values of $\theta$ in $[0, 2\pi)$ is $\sec\theta$ undefined? Explain why in terms of the unit circle. (2 marks)
Comprehensive answers (click to reveal)
Activity 1: 1. $\sec\theta = 13/5$; $\sin\theta = -\sqrt{1-25/169} = -12/13$; $\csc\theta = -13/12$ · 2. $\cot(\pi/6) = (\sqrt{3}/2)/(1/2) = \sqrt{3}$ · 3. $\theta = \pi/2$ and $\theta = 3\pi/2$ · 4. $\sin(5\pi/6) = 1/2$, $\csc(5\pi/6) = 2$ · 5. $\cot\theta = \cos\theta/\sin\theta$ has $\sin\theta$ in the denominator, same as $\csc\theta = 1/\sin\theta$; both undefined when $\sin\theta = 0$, i.e. $\theta = n\pi$.
Odd one out: B is incorrect. $\sec\theta = 1/\cos\theta$ is undefined when $\cos\theta = 0$, which occurs at $\theta = \pi/2$ and $\theta = 3\pi/2$, not at $\theta = 0$ and $\theta = \pi$ (where $\cos\theta = \pm 1 \neq 0$).
Q1 (3 marks): $\sec\theta = 1/\cos\theta = 13/5$ [1]. $\sin\theta < 0$ with $\cos\theta > 0$ means Q4; $\sin\theta = -\sqrt{1-25/169} = -\sqrt{144/169} = -12/13$ [1]. $\csc\theta = 1/\sin\theta = -13/12$ [1].
Q2 (1 mark): $\cot(\pi/6) = \cos(\pi/6)/\sin(\pi/6) = (\sqrt{3}/2)/(1/2) = \sqrt{3}$ [1].
Q3 (2 marks): $\sec\theta$ undefined at $\theta = \pi/2$ and $\theta = 3\pi/2$ [1]. On the unit circle, the x-coordinate (= $\cos\theta$) equals zero at the top ($\pi/2$) and bottom ($3\pi/2$) of the circle, making the denominator of $\sec\theta = 1/\cos\theta$ equal to zero [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering reciprocal trig questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.