# Secant and Cosecant

Source: algebrica.org — CC BY-NC 4.0
https://algebrica.org/secant-and-cosecant/)

## Secant

Consider the [unit circle](../unit-circle.md) centered at the origin \\(\text{O} = (0,0)\\) with radius \\(1\\). Let \\(\theta\\) be an angle in standard position, and denote by \\(\text{P}\\) the point on the circle where the terminal side of \\(\theta\\) intersects it. Draw the tangent line to the circle at the point \\(\text{P}\\), and let \\(\text{S}\\) be the point where this tangent line meets the \\(x\\)-axis. The secant of the angle \\(\theta\\) is defined as the signed length of the segment \\(\overline{OS}\\), that is, the abscissa \\(x_S\\) of the point \\(\text{S}\\):

\\[
\sec(\theta) = \overline{OS} = x_S
\\]

To express this length in terms of familiar trigonometric quantities, consider the right triangle formed by \\(\text{O}\\), \\(\text{P}\\), and the foot of the perpendicular from \\(\text{P}\\) to the \\(x\\)-axis. Since \\(\text{OP} = 1\\) and the horizontal component of \\(\text{P}\\) is \\(\cos(\theta)\\), while the tangent at \\(\text{P}\\) is perpendicular to the radius \\(\overline{OP}\\), one can establish by similar triangles that:

\\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\\]

Since the secant is the reciprocal of the [cosine](../sine-and-cosine.md), it is defined only at angles where the cosine does not vanish. The cosine equals zero at all odd multiples of \\(\pi/2\\), so the domain of the secant excludes precisely those values:

\\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\qquad \forall\\, \theta \neq \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}
\\]

From the geometric construction, the secant measures the factor by which the unit radius must be extended to reach the point \\(\text{S}\\) where the tangent line at \\(\text{P}\\) meets the \\(x\\)-axis. This interpretation makes it evident why \\(|\sec(\theta)| \geq 1\\) wherever the function is defined: the intersection point \\(\text{S}\\) necessarily lies at a distance from the origin no smaller than the radius of the unit circle itself.

> This section examines the secant from a geometric point of view. For the analytical properties of the function, including domain, symmetry, limits, derivatives, and integrals, see the dedicated entry on the [secant function](../secant-function.md).

- - -
## Common values of the secant

Below are some commonly known values of \\(\sec(\theta)\\) for selected angles, useful in various applications of trigonometry:

\\[
\begin{align}
\theta &= 0^\circ = 0\\,\text{rad}      && \sec(\theta) = 1 \\\\[6pt]
\theta &= 30^\circ = \pi/6\\,\text{rad} && \sec(\theta) = \tfrac{2\sqrt{3}}{3} \\\\[6pt]
\theta &= 45^\circ = \pi/4\\,\text{rad} && \sec(\theta) = \sqrt{2} \\\\[6pt]
\theta &= 60^\circ = \pi/3\\,\text{rad} && \sec(\theta) = 2 \\\\[6pt]
\theta &= 90^\circ = \pi/2\\,\text{rad} && \sec(\theta) \text{ is undefined}
\end{align}
\\]

- - -
## Trigonometric identities for the secant

+ \\[ \text{1. } \quad \sec x = \frac{1}{\cos x} \\]

+ \\[ \text{2. } \quad 1 + \tan^{2} x = \sec^{2} x \\]

+ \\[ \text{3. } \quad \sec(-x) = \sec x \\]

+ \\[ \text{4. } \quad \sec x \\,\tan x = \frac{\sin x}{\cos^{2} x} \\]

+ \\[ \text{5. } \quad \sec^{2} x - \tan^{2} x = 1 \\]

> These formulas collect the most useful identities involving the secant, including the reciprocal definition, the Pythagorean identity, the symmetry relation, and common algebraic transformations. For a broader overview, refer to the full collection of [trigonometric identities](../trigonometric-identities.md).

- - -
## Cosecant

Consider again the same construction: the tangent line drawn at \\(\text{P}\\) to the unit circle meets the \\(y\\)-axis at a point \\(\text{Q}\\). The cosecant of the angle \\(\theta\\) is defined as the signed length of the segment \\(\overline{OQ}\\), that is, the ordinate \\(y_Q\\) of the point \\(\text{Q}\\):

\\[
\csc(\theta) = \overline{OQ} = y_Q
\\]

By an argument analogous to that given for the secant, applying similar triangles to the configuration yields the following expression in terms of the sine:

\\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\\]

Since the cosecant is the reciprocal of the [sine](../sine-and-cosine.md), it is defined only at angles where the sine does not vanish. The sine equals zero at all integer multiples of \\(\pi\\), so the domain of the cosecant excludes precisely those values:

\\[
\csc(\theta) = \frac{1}{\sin(\theta)} \qquad \forall\\, \theta \neq k\pi, \quad k \in \mathbb{Z}
\\]

Analogous to the secant, the cosecant measures the factor by which the unit radius must be extended to reach the point \\(\text{Q}\\) where the tangent line at \\(\text{P}\\) meets the \\(y\\)-axis. This interpretation makes it evident why \\(|\csc(\theta)| \geq 1\\) wherever the function is defined: the intersection point \\(\text{Q}\\) necessarily lies at a distance from the origin no smaller than the radius of the unit circle itself.

> This section examines the cosecant from a geometric point of view. For the analytical properties of the function, including domain, symmetry, limits, derivatives, and integrals, see the dedicated entry on the [cosecant function](../cosecant-function.md).

- - -
## Geometric interpretation

Both definitions stem from a single geometric object: the [tangent](../tangent-and-cotangent.md) line drawn at \\(\text{P}\\) simultaneously determines the point \\(\text{S}\\) on the \\(x\\)-axis and the point \\(\text{Q}\\) on the \\(y\\)-axis, yielding the secant and the cosecant from one construction.

This also makes transparent the asymmetric behaviour of the two functions. When the terminal side of \\(\theta\\) approaches a horizontal position, the tangent line at \\(\text{P}\\) becomes nearly parallel to the \\(x\\)-axis, driving \\(\text{S}\\) to infinity and making the secant unbounded, while \\(\text{Q}\\) remains well-defined. The situation is reversed when the terminal side approaches a vertical position.

- - -
## Common values of the cosecant

Below are some commonly known values of \\(\csc(\theta)\\) for selected angles, useful in various applications of trigonometry:

\\[
\begin{align}
\theta &= 0^\circ = 0\\,\text{rad}      && \csc(\theta) \text{ is undefined} \\\\[6pt]
\theta &= 30^\circ = \pi/6\\,\text{rad} && \csc(\theta) = 2 \\\\[6pt]
\theta &= 45^\circ = \pi/4\\,\text{rad} && \csc(\theta) = \sqrt{2} \\\\[6pt]
\theta &= 60^\circ = \pi/3\\,\text{rad} && \csc(\theta) = \tfrac{2\sqrt{3}}{3} \\\\[6pt]
\theta &= 90^\circ = \pi/2\\,\text{rad} && \csc(\theta) = 1
\end{align}
\\]

- - -
## Secant and cosecant functions

The [secant function](../secant-function.md) \\(f(x) = \sec(x)\\) assigns to each angle \\(x\\), measured in radians, the value \\(1/\cos(x)\\). Its graph is a periodic curve with period \\(2\pi\\) and features vertical [asymptotes](../asymptotes.md) at the points where the cosine vanishes, that is, at \\(x = \pi/2 + k\pi\\) for \\(k \in \mathbb{Z}\\). The [domain](../determining-the-domain-of-a-function.md) of \\(\sec(x)\\) consists of all real numbers except those points, while its range is \\((-\infty, -1] \cup [1, +\infty)\\).

+ Domain: \\( \{ x \in \mathbb{R} : \cos(x) \neq 0 \} = \{ x \in \mathbb{R} : x \neq \pi/2 + k\pi \text{ for all } k \in \mathbb{Z} \} \\)
+ Range: \\( y \in (-\infty, -1] \cup [1, \infty) \\)
+ Periodicity: periodic in \\( x \\) with period \\( 2\pi \\)
+ Parity: [even](../even-and-odd-functions.md), \\( \sec(-x) = \sec(x) \\)

---

The [cosecant function](../cosecant-function.md) \\(f(x) = \csc(x)\\) assigns to each angle \\(x\\), measured in radians, the value \\(1/\sin(x)\\). Its graph is a periodic curve with period \\(2\pi\\) and features vertical asymptotes at the points where the sine vanishes, that is, at \\(x = k\pi\\) for \\(k \in \mathbb{Z}\\). The [domain](../determining-the-domain-of-a-function.md) of \\(\csc(x)\\) consists of all real numbers except those points, while its range is \\((-\infty, -1] \cup [1, +\infty)\\).

+ Domain: \\( \{ x \in \mathbb{R} : \sin(x) \neq 0 \} = \{ x \in \mathbb{R} : x \neq k\pi \text{ for all } k \in \mathbb{Z} \} \\)
+ Range: \\( y \in (-\infty, -1] \cup [1, \infty) \\)
+ Periodicity: periodic in \\( x \\) with period \\( 2\pi \\)
+ Parity: [odd](../even-and-odd-functions.md), \\( \csc(-x) = -\csc(x) \\)

- - -
## Trigonometric identities for the cosecant

+ \\[ \text{1. } \quad \csc x = \frac{1}{\sin x} \\]

+ \\[ \text{2. } \quad 1 + \cot^{2} x = \csc^{2} x \\]

+ \\[ \text{3. } \quad \csc(-x) = -\\,\csc x \\]

+ \\[ \text{4. } \quad \csc x \\,\cot x = \frac{\cos x}{\sin^{2} x} \\]

+ \\[ \text{5. } \quad \csc^{2} x - \cot^{2} x = 1 \\]

> These formulas collect the most useful identities involving the cosecant, including the reciprocal definition, the Pythagorean identity, the symmetry relation, and common algebraic transformations. For a broader overview, refer to the full collection of [trigonometric identities](../trigonometric-identities.md).