Reciprocals of trigonometric functions
Get to grips with the reciprocal trigonometric functions — secant, cosecant and cotangent — for Year 11 Specialist Mathematics in Queensland (QCAA). Each one is simply a familiar ratio turned upside down: secant reciprocates cosine, cosecant reciprocates sine, and cotangent reciprocates tangent.
You will learn to define and use these functions, work out simplified exact values at the special angles, and recognise where each function is undefined — the groundwork for the Pythagorean identities and trigonometric modelling later in the course.
Theory
The reciprocal trigonometric functions — secant, cosecant and cotangent — are the reciprocals of cosine, sine and tangent in Year 11 Specialist Mathematics (QCAA, Queensland). This page defines them, shows how to read off simplified exact values, and where each one is undefined.
The three reciprocal trigonometric functions are built by turning each familiar ratio upside down. Secant is the reciprocal of cosine, cosecant is the reciprocal of sine, and cotangent is the reciprocal of tangent.
In symbols, \(\sec x=\dfrac{1}{\cos x}\), \(\operatorname{cosec} x=\dfrac{1}{\sin x}\) and \(\cot x=\dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}\). Australian courses write cosecant as cosec (the same function is written \(\csc\) elsewhere).
Because you cannot divide by zero, each reciprocal function is undefined wherever its base function equals zero: \(\sec x\) fails where \(\cos x=0\), while \(\operatorname{cosec} x\) and \(\cot x\) both fail where \(\sin x=0\).
To evaluate one at a special angle, find the exact value of the base ratio first, then take its reciprocal and rationalise any surd. Every answer stays exact — surds and \(\pi\) are kept, never turned into decimals.
The three definitions, each undefined where its denominator is zero:
The simplified exact values at the first-quadrant special angles (read a base ratio, then take its reciprocal):
| \(x\) | \(0\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{4}\) | \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{2}\) |
|---|---|---|---|---|---|
| \(\sec x\) | \(1\) | \(\dfrac{2\sqrt{3}}{3}\) | \(\sqrt{2}\) | \(2\) | undefined |
| \(\operatorname{cosec} x\) | undefined | \(2\) | \(\sqrt{2}\) | \(\dfrac{2\sqrt{3}}{3}\) | \(1\) |
| \(\cot x\) | undefined | \(\sqrt{3}\) | \(1\) | \(\dfrac{\sqrt{3}}{3}\) | \(0\) |
How to evaluate a reciprocal function
- Identify the base function: secant pairs with cosine, cosecant with sine, cotangent with tangent (or cosine over sine).
- Evaluate the base ratio at the angle, using the special triangles or the unit circle, and keep the correct sign for the quadrant.
- Reciprocate: take one over that value; if the base ratio is \(0\), the reciprocal is undefined.
- Rationalise any surd denominator so the exact value is in simplest form.
Secant is the reciprocal of cosine; evaluate the cosine, then flip it:
| \(\sec\dfrac{\pi}{3}\) | \(=\) | \(\dfrac{1}{\cos\dfrac{\pi}{3}}\) |
| \(\cos\dfrac{\pi}{3}\) | \(=\) | \(\dfrac{1}{2}\) |
| \(=\) | \(\dfrac{1}{\dfrac{1}{2}}\) | |
| \(=\) | \(2\) |
\(\sec\dfrac{\pi}{3}=2\).
Cosecant is the reciprocal of sine; flip, then rationalise the surd denominator:
| \(\operatorname{cosec}\dfrac{\pi}{3}\) | \(=\) | \(\dfrac{1}{\sin\dfrac{\pi}{3}}\) |
| \(\sin\dfrac{\pi}{3}\) | \(=\) | \(\dfrac{\sqrt{3}}{2}\) |
| \(=\) | \(\dfrac{2}{\sqrt{3}}\) | |
| \(=\) | \(\dfrac{2\sqrt{3}}{3}\) |
\(\operatorname{cosec}\dfrac{\pi}{3}=\dfrac{2\sqrt{3}}{3}\).
The angle \(\dfrac{2\pi}{3}\) is in the second quadrant, where cosine is negative. Find the cosine, then take its reciprocal:
| \(\cos\dfrac{2\pi}{3}\) | \(=\) | \(-\dfrac{1}{2}\) |
| \(\sec\dfrac{2\pi}{3}\) | \(=\) | \(\dfrac{1}{\cos\dfrac{2\pi}{3}}\) |
| \(=\) | \(\dfrac{1}{-\dfrac{1}{2}}\) | |
| \(=\) | \(-2\) |
\(\sec\dfrac{2\pi}{3}=-2\).
Write each function as a reciprocal, then divide the fractions:
| \(\dfrac{\sec x}{\operatorname{cosec} x}\) | \(=\) | \(\dfrac{\dfrac{1}{\cos x}}{\dfrac{1}{\sin x}}\) |
| \(=\) | \(\dfrac{1}{\cos x}\times\dfrac{\sin x}{1}\) | |
| \(=\) | \(\dfrac{\sin x}{\cos x}\) | |
| \(=\) | \(\tan x\) |
\(\dfrac{\sec x}{\operatorname{cosec} x}=\tan x\).
Common pitfalls
Frequently asked questions
What are sec, cosec and cot?
They are the reciprocal trigonometric functions: \(\sec x=\dfrac{1}{\cos x}\), \(\operatorname{cosec} x=\dfrac{1}{\sin x}\) and \(\cot x=\dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}\).
Is cosec the reciprocal of sine or cosine?
Cosecant is the reciprocal of sine: \(\operatorname{cosec} x=\dfrac{1}{\sin x}\). Secant is the reciprocal of cosine. The names deliberately do not line up with the ratios.
Where is sec x undefined?
Wherever \(\cos x=0\), because you cannot divide by zero. On \(0\le x\le 2\pi\) that is \(x=\dfrac{\pi}{2}\) and \(x=\dfrac{3\pi}{2}\).
How do I find sec of pi on 3?
Take the reciprocal of \(\cos\dfrac{\pi}{3}=\dfrac{1}{2}\), giving \(\sec\dfrac{\pi}{3}=\dfrac{1}{1/2}=2\).
Is sec x the same as cos inverse x?
No. \(\sec x=\dfrac{1}{\cos x}\) is a reciprocal, while \(\cos^{-1}x\) is the inverse cosine that returns an angle. They are different functions.
How do I get cot from cos and sin?
Cotangent is cosine divided by sine: \(\cot x=\dfrac{\cos x}{\sin x}\), which is also \(\dfrac{1}{\tan x}\).