Trigonometric Functions of a Real Number (Unit Circle Approach)

Defining Trigonometric Functions using the Unit Circle

Up to this point, we have primarily defined trigonometric ratios for acute angles ($0^\circ$ to $90^\circ$) within the context of right-angled triangles. However, trigonometric functions are defined for any real number, representing angles beyond the $0^\circ$ to $90^\circ$ range, including negative angles and angles greater than $360^\circ$. The most powerful way to extend the definition of trigonometric ratios to encompass all angles is by using the Unit Circle.

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The Unit Circle

The unit circle is a circle centered at the origin (0, 0) of the Cartesian coordinate system with a radius of 1 unit. The equation of the unit circle is $ x^2 + y^2 = 1 $. Every point P(x, y) on the unit circle satisfies this equation.

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Defining Trigonometric Functions

To define the trigonometric functions for any angle using the unit circle, we follow these steps:

1. Place the Angle in Standard Position: Consider any angle $\theta$. We place this angle in standard position on the coordinate plane. The vertex of the angle is at the origin (0,0). The initial side of the angle lies along the positive x-axis.

2. Identify the Intersection Point: The terminal side of the angle $\theta$ is obtained by rotating the initial side by the measure of $\theta$ (anticlockwise for positive $\theta$, clockwise for negative $\theta$). This terminal side will intersect the unit circle at exactly one point. Let the coordinates of this point be $P(x, y)$.

3. Define the Trigonometric Functions: The six trigonometric functions of the angle $\theta$ are defined in terms of the coordinates $x$ and $y$ of this point $P(x, y)$ on the unit circle:

   • Cosine Function ($\cos \theta$): The cosine of the angle $\theta$ is defined as the x-coordinate of the point $P(x, y)$ where the terminal side of $\theta$ intersects the unit circle.

     $\mathbf{\cos \theta = x}$

   • Sine Function ($\sin \theta$): The sine of the angle $\theta$ is defined as the y-coordinate of the point $P(x, y)$ where the terminal side of $\theta$ intersects the unit circle.

     $\mathbf{\sin \theta = y}$

   • Tangent Function ($\tan \theta$): The tangent of the angle $\theta$ is defined as the ratio of the y-coordinate to the x-coordinate of the point $P(x, y)$.

     $\mathbf{\tan \theta = \frac{y}{x}}$, provided $x \ne 0$.

   • Cosecant Function ($\text{cosec} \, \theta$ or $\csc \theta$): The cosecant of the angle $\theta$ is defined as the reciprocal of the y-coordinate of the point $P(x, y)$.

     $\mathbf{\text{cosec} \, \theta = \frac{1}{y}}$, provided $y \ne 0$.

   • Secant Function ($\sec \theta$): The secant of the angle $\theta$ is defined as the reciprocal of the x-coordinate of the point $P(x, y)$.

     $\mathbf{\sec \theta = \frac{1}{x}}$, provided $x \ne 0$.

   • Cotangent Function ($\cot \theta$): The cotangent of the angle $\theta$ is defined as the ratio of the x-coordinate to the y-coordinate of the point $P(x, y)$.

     $\mathbf{\cot \theta = \frac{x}{y}}$, provided $y \ne 0$.

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Consistency with Right Triangle Definitions (for Acute Angles)

Let's see how these unit circle definitions are consistent with the right-angled triangle definitions for acute angles ($0^\circ < \theta < 90^\circ$).

If $\theta$ is an acute angle, the terminal side lies in the first quadrant. The point $P(x, y)$ on the unit circle forms a right-angled triangle with the origin (0, 0) and the point $Q(x, 0)$ on the x-axis. In this right triangle $\triangle OQP$:

• The angle at O is $\theta$.

• The side OQ (adjacent to $\theta$) has length $|x| = x$ (since x > 0 in the first quadrant).

• The side PQ (opposite to $\theta$) has length $|y| = y$ (since y > 0 in the first quadrant).

• The hypotenuse OP is the radius of the unit circle, so its length is 1.

Now, apply the right triangle definitions:

$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{PQ}{OP} = \frac{y}{1} = y $

$ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{OQ}{OP} = \frac{x}{1} = x $

These match the unit circle definitions $\sin \theta = y$ and $\cos \theta = x$. The other ratios also follow:

$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{y}{x} $

$ \text{cosec} \, \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{y} $

$ \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{x} $

$ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{x}{y} $

The unit circle definition thus provides a consistent and extended framework for trigonometric functions for all angles.

Angle as a Real Number

When angle $\theta$ is measured in radians, the arc length along the unit circle from (1, 0) to P(x, y) is numerically equal to $\theta$ (since $s = r\theta$ and $r=1$). This means the trigonometric functions can be viewed as functions of a real number (the radian measure), not just an angle.

Example 1: Finding Trigonometric Values using the Unit Circle

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Domain and Range of Trigonometric Functions

The unit circle definition allows us to determine the set of all possible input values (the domain) and the set of all possible output values (the range) for each of the six trigonometric functions.

The domain is the set of all angles $\theta$ for which the function is defined based on the coordinates $(x, y)$ of the point $P$ where the terminal side of $\theta$ intersects the unit circle. The range is the set of all possible values that the function can output.

Recall the definitions: $ \cos \theta = x $, $ \sin \theta = y $, $ \tan \theta = y/x $, $ \text{cosec} \, \theta = 1/y $, $ \sec \theta = 1/x $, $ \cot \theta = x/y $, where P(x, y) is on the unit circle.

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Sine Function ($f(\theta) = \sin \theta$)

• Definition: $ \sin \theta = y $ (the y-coordinate of P).

• Domain: For any angle $\theta$, whether positive, negative, or zero, the terminal side will intersect the unit circle at exactly one point P(x, y). The y-coordinate $y$ is always well-defined for any angle $\theta$. Therefore, the domain of $\sin \theta$ is the set of all real numbers, denoted by $ \mathbb{R} $.

  Domain of $\sin \theta$: $ \mathbb{R} $

• Range: As the point P moves around the unit circle, its y-coordinate ($y = \sin \theta$) varies between a minimum value of -1 (at $\theta = 270^\circ, -90^\circ$, etc.) and a maximum value of 1 (at $\theta = 90^\circ, 450^\circ$, etc.). All values between -1 and 1 (inclusive) are possible y-coordinates for points on the unit circle. Therefore, the range of $\sin \theta$ is the closed interval $[-1, 1]$.

  Range of $\sin \theta$: $ [-1, 1] $

  $-1 \leq \sin \theta \leq 1$

Cosine Function ($f(\theta) = \cos \theta$)

• Definition: $ \cos \theta = x $ (the x-coordinate of P).

• Domain: Similar to the sine function, for any angle $\theta$, the terminal side intersects the unit circle at a point P(x, y), and the x-coordinate $x$ is always well-defined. Therefore, the domain of $\cos \theta$ is the set of all real numbers, $ \mathbb{R} $.

  Domain of $\cos \theta$: $ \mathbb{R} $

• Range: As the point P moves around the unit circle, its x-coordinate ($x = \cos \theta$) varies between a minimum value of -1 (at $\theta = 180^\circ, 540^\circ$, etc.) and a maximum value of 1 (at $\theta = 0^\circ, 360^\circ$, etc.). All values between -1 and 1 (inclusive) are possible x-coordinates for points on the unit circle. Therefore, the range of $\cos \theta$ is the closed interval $[-1, 1]$.

  Range of $\cos \theta$: $ [-1, 1] $

  $-1 \leq \cos \theta \leq 1$

Tangent Function ($f(\theta) = \tan \theta$)

• Definition: $ \tan \theta = y/x $.

• Domain: The tangent function is defined only when the denominator $x \ne 0$. The x-coordinate of a point on the unit circle is zero when the point is on the y-axis. These points are (0, 1) and (0, -1). The angles corresponding to these points are $\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots$ (in the anticlockwise direction) and $ -\frac{\pi}{2}, -\frac{3\pi}{2}, \dots $ (in the clockwise direction). These angles can be represented by the general form $ \frac{\pi}{2} + n\pi $, where $n$ is any integer ($n \in \mathbb{Z}$). Therefore, the domain of $\tan \theta$ is the set of all real numbers except these angles.

  Domain of $\tan \theta$: $ \mathbb{R} - \left\{ \theta \mid \cos \theta = 0 \right\} = \mathbb{R} - \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} $

• Range: As the terminal side of $\theta$ rotates, the ratio $y/x$ can take any real value. Consider angles close to $\pi/2$. As $\theta$ approaches $\pi/2$ from the left (e.g., $89^\circ$), $y$ is close to 1 and $x$ is a small positive number, making $y/x$ a large positive number. As $\theta$ approaches $\pi/2$ from the right (e.g., $91^\circ$), $y$ is close to 1 and $x$ is a small negative number, making $y/x$ a large negative number. The ratio can take any real value. Therefore, the range of $\tan \theta$ is the set of all real numbers, $ \mathbb{R} $.

  Range of $\tan \theta$: $ \mathbb{R} $

Cosecant Function ($f(\theta) = \text{cosec} \, \theta$)

• Definition: $ \text{cosec} \, \theta = 1/y $.

• Domain: The cosecant function is defined only when the denominator $y \ne 0$. The y-coordinate of a point on the unit circle is zero when the point is on the x-axis. These points are (1, 0) and (-1, 0). The angles corresponding to these points are $\theta = 0, \pi, 2\pi, 3\pi, \dots$ and $ -\pi, -2\pi, \dots $. These angles can be represented by the general form $ n\pi $, where $n$ is any integer ($n \in \mathbb{Z}$). Therefore, the domain of $\text{cosec} \, \theta$ is the set of all real numbers except these angles.

  Domain of $\text{cosec} \, \theta$: $ \mathbb{R} - \left\{ \theta \mid \sin \theta = 0 \right\} = \mathbb{R} - \left\{ n\pi \mid n \in \mathbb{Z} \right\} $

• Range: Since $-1 \leq y \leq 1$ for points on the unit circle, and $y \ne 0$ for $\text{cosec} \, \theta$ to be defined, the values of $y$ are restricted to $[-1, 0) \cup (0, 1]$. The reciprocal $1/y$ will therefore have values $\le -1$ (when $-1 \le y < 0$) or $\ge 1$ (when $0 < y \le 1$). Thus, the range of $\text{cosec} \, \theta$ is $(-\infty, -1] \cup [1, \infty)$.

  Range of $\text{cosec} \, \theta$: $ (-\infty, -1] \cup [1, \infty) $

  $\text{cosec} \, \theta \leq -1$ or $\text{cosec} \, \theta \geq 1$

Secant Function ($f(\theta) = \sec \theta$)

• Definition: $ \sec \theta = 1/x $.

• Domain: The secant function is defined only when the denominator $x \ne 0$. This is the same condition as for the tangent function. Therefore, the domain of $\sec \theta$ is the set of all real numbers except angles of the form $ \frac{\pi}{2} + n\pi $, where $n \in \mathbb{Z} $.

  Domain of $\sec \theta$: $ \mathbb{R} - \left\{ \theta \mid \cos \theta = 0 \right\} = \mathbb{R} - \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} $

• Range: Similar to the cosecant function, since $-1 \leq x \leq 1$ for points on the unit circle, and $x \ne 0$ for $\sec \theta$ to be defined, the values of $x$ are restricted to $[-1, 0) \cup (0, 1]$. The reciprocal $1/x$ will therefore have values $\le -1$ (when $-1 \le x < 0$) or $\ge 1$ (when $0 < x \le 1$). Thus, the range of $\sec \theta$ is $(-\infty, -1] \cup [1, \infty)$.

  Range of $\sec \theta$: $ (-\infty, -1] \cup [1, \infty) $

  $\sec \theta \leq -1$ or $\sec \theta \geq 1$

Cotangent Function ($f(\theta) = \cot \theta$)

• Definition: $ \cot \theta = x/y $.

• Domain: The cotangent function is defined only when the denominator $y \ne 0$. This is the same condition as for the cosecant function. Therefore, the domain of $\cot \theta$ is the set of all real numbers except angles of the form $ n\pi $, where $n \in \mathbb{Z} $.

  Domain of $\cot \theta$: $ \mathbb{R} - \left\{ \theta \mid \sin \theta = 0 \right\} = \mathbb{R} - \left\{ n\pi \mid n \in \mathbb{Z} \right\} $

• Range: As the terminal side of $\theta$ rotates, the ratio $x/y$ can take any real value. Consider angles close to $0$. As $\theta$ approaches $0$ from above, $x$ is close to 1 and $y$ is a small positive number, making $x/y$ a large positive number. As $\theta$ approaches $0$ from below, $x$ is close to 1 and $y$ is a small negative number, making $x/y$ a large negative number. The ratio can take any real value. Therefore, the range of $\cot \theta$ is the set of all real numbers, $ \mathbb{R} $.

  Range of $\cot \theta$: $ \mathbb{R} $

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Summary Table: Domain and Range of Trigonometric Functions

The table below summarises the domain and range for each of the six trigonometric functions:

Function	Domain	Range
$y = \sin \theta$	$ \mathbb{R} $	$ [-1, 1] $
$y = \cos \theta$	$ \mathbb{R} $	$ [-1, 1] $
$y = \tan \theta$	$ \mathbb{R} - \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} $	$ \mathbb{R} $
$y = \text{cosec} \, \theta$	$ \mathbb{R} - \left\{ n\pi \mid n \in \mathbb{Z} \right\} $	$ (-\infty, -1] \cup [1, \infty) $
$y = \sec \theta$	$ \mathbb{R} - \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} $	$ (-\infty, -1] \cup [1, \infty) $
$y = \cot \theta$	$ \mathbb{R} - \left\{ n\pi \mid n \in \mathbb{Z} \right\} $	$ \mathbb{R} $

Understanding the domain and range is critical for analysing the behaviour of trigonometric functions and for solving equations and inequalities involving them.

Signs of Trigonometric Functions in Different Quadrants

Using the unit circle definition of trigonometric functions (where $ \cos \theta = x $ and $ \sin \theta = y $ for a point P(x, y) on the unit circle corresponding to angle $\theta$), we can determine the sign (positive or negative) of each trigonometric function based on the quadrant in which the terminal side of the angle $\theta$ lies.

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Signs of Coordinates in Quadrants

The Cartesian coordinate plane is divided into four quadrants by the x and y axes. The signs of the x and y coordinates of a point depend on which quadrant the point lies in.

• Quadrant I: This quadrant is where both x and y coordinates are positive. It corresponds to angles $ \theta $ such that $ 0 < \theta < \frac{\pi}{2} $ (or $ 0^\circ < \theta < 90^\circ $).

  In Quadrant I: $ x > 0, y > 0 $.

• Quadrant II: This quadrant is where the x coordinate is negative and the y coordinate is positive. It corresponds to angles $ \theta $ such that $ \frac{\pi}{2} < \theta < \pi $ (or $ 90^\circ < \theta < 180^\circ $).

  In Quadrant II: $ x < 0, y > 0 $.

• Quadrant III: This quadrant is where both x and y coordinates are negative. It corresponds to angles $ \theta $ such that $ \pi < \theta < \frac{3\pi}{2} $ (or $ 180^\circ < \theta < 270^\circ $).

  In Quadrant III: $ x < 0, y < 0 $.

• Quadrant IV: This quadrant is where the x coordinate is positive and the y coordinate is negative. It corresponds to angles $ \theta $ such that $ \frac{3\pi}{2} < \theta < 2\pi $ (or $ 270^\circ < \theta < 360^\circ $).

  In Quadrant IV: $ x > 0, y < 0 $.

Angles that fall exactly on the axes (like $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$, etc.) are called quadrantal angles and do not lie "in" a quadrant. Their trigonometric values are $0, \pm 1,$ or undefined.

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Signs of Trigonometric Functions in Each Quadrant

Now, we can determine the sign of each trigonometric function in each quadrant by examining the definitions and the signs of $x$ and $y$ in that quadrant.

• Quadrant I ($x > 0, y > 0$):

  • $\sin \theta = y $ is positive (+)

  • $\cos \theta = x $ is positive (+)

  • $\tan \theta = y/x $ is $(+)/(+) = $ positive (+)

  • $\text{cosec} \, \theta = 1/y $ is $1/(+) = $ positive (+)

  • $\sec \theta = 1/x $ is $1/(+) = $ positive (+)

  • $\cot \theta = x/y $ is $(+)/(+) = $ positive (+)

  Conclusion: All trigonometric functions are positive in Quadrant I.

• Quadrant II ($x < 0, y > 0$):

  • $\sin \theta = y $ is positive (+)

  • $\cos \theta = x $ is negative (-)

  • $\tan \theta = y/x $ is $(+)/(-) = $ negative (-)

  • $\text{cosec} \, \theta = 1/y $ is $1/(+) = $ positive (+)

  • $\sec \theta = 1/x $ is $1/(-) = $ negative (-)

  • $\cot \theta = x/y $ is $(-)/(+) = $ negative (-)

  Conclusion: Only $\sin \theta$ and its reciprocal $\text{cosec} \, \theta$ are positive in Quadrant II.

• Quadrant III ($x < 0, y < 0$):

  • $\sin \theta = y $ is negative (-)

  • $\cos \theta = x $ is negative (-)

  • $\tan \theta = y/x $ is $(-)/(-) = $ positive (+)

  • $\text{cosec} \, \theta = 1/y $ is $1/(-) = $ negative (-)

  • $\sec \theta = 1/x $ is $1/(-) = $ negative (-)

  • $\cot \theta = x/y $ is $(-)/(-) = $ positive (+)

  Conclusion: Only $\tan \theta$ and its reciprocal $\cot \theta$ are positive in Quadrant III.

• Quadrant IV ($x > 0, y < 0$):

  • $\sin \theta = y $ is negative (-)

  • $\cos \theta = x $ is positive (+)

  • $\tan \theta = y/x $ is $(+)/(-) = $ negative (-)

  • $\text{cosec} \, \theta = 1/y $ is $1/(-) = $ negative (-)

  • $\sec \theta = 1/x $ is $1/(+) = $ positive (+)

  • $\cot \theta = x/y $ is $(+)/(-) = $ negative (-)

  Conclusion: Only $\cos \theta$ and its reciprocal $\sec \theta$ are positive in Quadrant IV.

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Mnemonic: The ASTC Rule (or CAST Rule)

To easily remember which trigonometric functions are positive in each quadrant, we use a simple mnemonic. Starting from Quadrant I and moving counter-clockwise (in the direction of positive angles), the letters A, S, T, C stand for the functions that are positive in that quadrant:

• A in Quadrant I: All functions are positive.

• S in Quadrant II: Only Sine (and its reciprocal, cosecant) is positive.

• T in Quadrant III: Only Tangent (and its reciprocal, cotangent) is positive.

• C in Quadrant IV: Only Cosine (and its reciprocal, secant) is positive.

Common mnemonics to remember the order ASTC (starting from Q1 counter-clockwise):

• All Students Take Calculus

• Add Sugar To Coffee

• All Silver Tea Cups

Sometimes this rule is presented starting from Quadrant IV and moving clockwise, using the word "CAST". In this case, C is in Q4, A in Q1, S in Q2, T in Q3. Both mnemonics convey the same information.

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Examples

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Note for Competitive Exams

Periodicity of Trigonometric Functions

One of the key properties of trigonometric functions is their periodicity. This means that their values repeat at regular intervals as the angle changes. This repeating nature is a direct consequence of how angles are represented on the unit circle.

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Definition of a Periodic Function

A function $f(x)$ is called a periodic function if there exists a positive real number $P$ such that $f(x + P) = f(x)$ for all values of $x$ in the domain of $f$. The smallest such positive number $P$ is called the fundamental period or simply the period of the function.

Essentially, a periodic function traces out the same shape over and over again over intervals of length equal to its period.

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Periodicity from the Unit Circle

Consider an angle $\theta$ in standard position on the unit circle, with its terminal side intersecting the circle at point P(x, y).

If we increase the angle $\theta$ by $2\pi$ radians (or $360^\circ$), the terminal side completes one full revolution and returns to the exact same position as for angle $\theta$. This means the terminal side of angle $\theta + 2\pi$ intersects the unit circle at the same point P(x, y).

Similarly, if we increase the angle by any integer multiple of $2\pi$ (e.g., $\theta + 4\pi, \theta - 2\pi, \theta + 2n\pi$ for $n \in \mathbb{Z}$), the terminal side ends up in the same position, intersecting the unit circle at the same point P(x, y).

Since the sine and cosine functions are defined directly by the y and x coordinates of the point P, their values will repeat every $2\pi$ radians.

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Periods of Sine, Cosine, Cosecant, and Secant Functions

• Sine Function:

  $ \sin \theta = y $

  Since the point P(x, y) for angle $\theta$ is the same as for $\theta + 2\pi$, the y-coordinate is the same:

  $\sin (\theta + 2\pi) = y = \sin \theta$

  More generally, $ \sin (\theta + 2n\pi) = \sin \theta $ for any integer $n$. The smallest positive value of $P$ for which $ \sin (\theta + P) = \sin \theta $ is $2\pi$.

  The fundamental period of $\sin \theta$ is $2\pi$.

• Cosine Function:

  $ \cos \theta = x $

  Since the point P(x, y) for angle $\theta$ is the same as for $\theta + 2\pi$, the x-coordinate is the same:

  $\cos (\theta + 2\pi) = x = \cos \theta$

  More generally, $ \cos (\theta + 2n\pi) = \cos \theta $ for any integer $n$. The smallest positive value of $P$ for which $ \cos (\theta + P) = \cos \theta $ is $2\pi$.

  The fundamental period of $\cos \theta$ is $2\pi$.

• Cosecant Function:

  $ \text{cosec} \, \theta = 1/y $ (defined for $y \ne 0$)

  Since $y$ repeats every $2\pi$, its reciprocal $1/y$ also repeats every $2\pi$ (where defined):

  $\text{cosec} \, (\theta + 2\pi) = \frac{1}{y} = \text{cosec} \, \theta$

  The smallest positive value of $P$ is $2\pi$.

  The fundamental period of $\text{cosec} \, \theta$ is $2\pi$.

• Secant Function:

  $ \sec \theta = 1/x $ (defined for $x \ne 0$)

  Since $x$ repeats every $2\pi$, its reciprocal $1/x$ also repeats every $2\pi$ (where defined):

  $\sec (\theta + 2\pi) = \frac{1}{x} = \sec \theta$

  The smallest positive value of $P$ is $2\pi$.

  The fundamental period of $\sec \theta$ is $2\pi$.

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Periods of Tangent and Cotangent Functions

• Tangent Function:

  $ \tan \theta = y/x $ (defined for $x \ne 0$)

  Consider adding $\pi$ radians ($180^\circ$) to the angle $\theta$. The terminal side of $\theta + \pi$ is diametrically opposite to the terminal side of $\theta$. If the point for $\theta$ is P(x, y), the point for $\theta + \pi$ is P'(-x, -y).

  

  Now evaluate $\tan(\theta + \pi)$ using the coordinates of P'(-x, -y):

  $ \tan (\theta + \pi) = \frac{\text{new y-coordinate}}{\text{new x-coordinate}} = \frac{-y}{-x} = \frac{y}{x} $

  Since $ \frac{y}{x} = \tan \theta $, we have:

  $\tan (\theta + \pi) = \tan \theta$

  This shows that the value of $\tan \theta$ repeats every $\pi$ radians. The smallest positive value of $P$ for which $ \tan (\theta + P) = \tan \theta $ is $\pi$.

  The fundamental period of $\tan \theta$ is $\pi$.

• Cotangent Function:

  $ \cot \theta = x/y $ (defined for $y \ne 0$)

  Using the same points P(x, y) and P'(-x, -y) for angles $\theta$ and $\theta + \pi$:

  $ \cot (\theta + \pi) = \frac{\text{new x-coordinate}}{\text{new y-coordinate}} = \frac{-x}{-y} = \frac{x}{y} $

  Since $ \frac{x}{y} = \cot \theta $, we have:

  $\cot (\theta + \pi) = \cot \theta$

  This shows that the value of $\cot \theta$ repeats every $\pi$ radians. The smallest positive value of $P$ for which $ \cot (\theta + P) = \cot \theta $ is $\pi$.

  The fundamental period of $\cot \theta$ is $\pi$.

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Summary of Fundamental Periods

The fundamental periods of the six trigonometric functions are:

Function	Fundamental Period
$\sin \theta$	$2\pi$ (or $360^\circ$)
$\cos \theta$	$2\pi$ (or $360^\circ$)
$\tan \theta$	$\pi$ (or $180^\circ$)
$\text{cosec} \, \theta$	$2\pi$ (or $360^\circ$)
$\sec \theta$	$2\pi$ (or $360^\circ$)
$\cot \theta$	$\pi$ (or $180^\circ$)

Understanding periodicity is crucial for sketching the graphs of trigonometric functions, solving trigonometric equations (which often have infinitely many solutions due to periodicity), and analysing phenomena in science and engineering that exhibit cyclical behaviour.

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Examples

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Note for Competitive Exams