Cosecant, Secant, and Cotangent

Three More Functions

Each of the three trig functions has a reciprocal:

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

No new geometry. Just “flip the fraction.”

From the Triangle

If sin⁡=oppositehypotenuse\sin = \frac{\text{opposite}}{\text{hypotenuse}}, then csc⁡\csc flips it:

FunctionRatio
csc⁡θ\csc \thetahypotenuse / opposite
sec⁡θ\sec \thetahypotenuse / adjacent
cot⁡θ\cot \thetaadjacent / opposite

On the Unit Circle

The hypotenuse is 1, so:

  • csc⁡θ=1y\csc \theta = \frac{1}{y}
  • sec⁡θ=1x\sec \theta = \frac{1}{x}
  • cot⁡θ=xy\cot \theta = \frac{x}{y}

When They’re Undefined

Each one breaks when its denominator is zero:

FunctionUndefined whenWhich means
csc⁡θ\csc \thetasin⁡θ=0\sin \theta = 0θ=0°,180°,360°,…\theta = 0°, 180°, 360°, \ldots
sec⁡θ\sec \thetacos⁡θ=0\cos \theta = 0θ=90°,270°,…\theta = 90°, 270°, \ldots
cot⁡θ\cot \thetasin⁡θ=0\sin \theta = 0θ=0°,180°,360°,…\theta = 0°, 180°, 360°, \ldots

These are the points where the unit circle crosses an axis, making xx or yy equal to zero.

Why Do They Exist?

Historically, having names for 1/sin⁡1/\sin and 1/cos⁡1/\cos saved a lot of writing.

Today, sin⁡\sin, cos⁡\cos, and tan⁡\tan do most of the work. But the reciprocal functions still show up in calculus and in certain identities, so they’re worth knowing.