The Cosecant Function | Moosa Academy

Beside sine, cosine and tangent sit their three reciprocals: cosecant, secant and cotangent. This lesson looks closely at the first of them, $\csc\theta$ — the reciprocal of the sine.

Concept The three reciprocal functions

\csc\theta = \dfrac{1}{\sin\theta}

\sec\theta = \dfrac{1}{\cos\theta}

\cot\theta = \dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta}

Each is a fraction, so each one breaks down wherever its denominator hits zero. Dividing by a value that shrinks toward 0 sends the result toward ±∞ — which is exactly what draws the vertical asymptotes on their graphs.

Graph csc θ against sin θ

Between 0° and 180° the branch opens upward, with its lowest point 1 at 90° — exactly where the sine reaches its maximum. Between 180° and 360° the branch opens downward, touching −1 at 270°. As $\sin\theta$ approaches 0, $\csc\theta$ races off to ±∞.

Note Properties of csc θ

Domain: every angle except the multiples of 180°, where $\sin\theta = 0$ .

Range: $\csc\theta \le -1$ or $\csc\theta \ge 1$ — never a value strictly between −1 and 1, because $|\sin\theta| \le 1$ means its reciprocal is at least 1 in size.

Period: 360°, inherited from the sine.

Example Evaluating the reciprocal

\csc 30° = \dfrac{1}{\sin 30°} = \dfrac{1}{0.5}

⟹ csc 30° = 2

\csc 90° = \dfrac{1}{\sin 90°} = \dfrac{1}{1}

⟹ csc 90° = 1, the smallest positive value it takes

\csc 180° = \dfrac{1}{\sin 180°} = \dfrac{1}{0}

⟹ undefined — an asymptote sits at 180°

Reference The three reciprocals compared

Property	csc θ	sec θ	cot θ
Definition	1 ÷ sin θ	1 ÷ cos θ	1 ÷ tan θ
Period	360°	360°	180°
Range	≤ −1 or ≥ 1	≤ −1 or ≥ 1	all real numbers
Asymptotes at	sin θ = 0	cos θ = 0	sin θ = 0

Summary

csc θ = 1 ÷ sin θ, one of the three reciprocal functions alongside sec and cot.
It is undefined at multiples of 180°, and those angles become vertical asymptotes.
Its values never fall strictly between −1 and 1; its period is 360°.
It touches the sine curve at the sine's extremes: 1 at 90° and −1 at 270°.