Graphs of Trigonometric Functions

Graph of Sine Function ($y = \sin x$)

Understanding the graphs of trigonometric functions is crucial for visualising their behaviour, identifying their properties (like domain, range, and periodicity), and solving equations and inequalities. The graph of the sine function, $y = \sin x$, is one of the most fundamental trigonometric graphs.

We can construct the graph by plotting points corresponding to various angles $x$ (often measured in radians) and their sine values, $y = \sin x$. The value of $\sin x$ for an angle $x$ is given by the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

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Key Points and Shape

Let's consider the values of $\sin x$ at some specific angles within one complete cycle, typically from $x = 0$ to $x = 2\pi$ radians ($0^\circ$ to $360^\circ$).

• At $x = 0$ ($0^\circ$): The terminal side is at (1, 0) on the unit circle. The y-coordinate is 0. So, $ \sin 0 = 0 $. This gives the point $(0, 0)$ on the graph.

• At $x = \frac{\pi}{2}$ ($90^\circ$): The terminal side is at (0, 1). The y-coordinate is 1. So, $ \sin (\frac{\pi}{2}) = 1 $. This gives the point $(\frac{\pi}{2}, 1)$ on the graph. This is a maximum value.

• At $x = \pi$ ($180^\circ$): The terminal side is at (-1, 0). The y-coordinate is 0. So, $ \sin \pi = 0 $. This gives the point $(\pi, 0)$ on the graph.

• At $x = \frac{3\pi}{2}$ ($270^\circ$): The terminal side is at (0, -1). The y-coordinate is -1. So, $ \sin (\frac{3\pi}{2}) = -1 $. This gives the point $(\frac{3\pi}{2}, -1)$ on the graph. This is a minimum value.

• At $x = 2\pi$ ($360^\circ$): The terminal side is back at (1, 0). The y-coordinate is 0. So, $ \sin (2\pi) = 0 $. This gives the point $(2\pi, 0)$ on the graph. This completes one cycle, and the values will repeat from here due to the periodicity of the sine function.

Plotting these key points $(0,0), (\frac{\pi}{2}, 1), (\pi, 0), (\frac{3\pi}{2}, -1), (2\pi, 0)$ and considering the smooth variation of the y-coordinate as the angle rotates around the unit circle, we obtain a characteristic wave-like curve.

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Graph of $y = \sin x$

The graph of $y = \sin x$ is a continuous wave that extends infinitely in both the positive and negative x-directions, repeating the same pattern over intervals of $2\pi$. This repeating pattern is called a sinusoid or sine wave.

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Properties of $y = \sin x$ from the Graph

We can infer several properties of the sine function by looking at its graph:

• Domain: The graph exists for all real values of $x$. Therefore, the domain is the set of all real numbers, $ \mathbb{R} $.

• Range: The minimum value of $y$ is -1 and the maximum value is 1. The graph never goes below -1 or above 1. Therefore, the range is the closed interval $ [-1, 1] $.

  $-1 \leq \sin x \leq 1$

• Periodicity: The graph repeats its complete cycle every $2\pi$ units along the x-axis. The fundamental period is $ 2\pi $.

  $\sin(x + 2\pi) = \sin x$

• Continuity: The graph is a smooth, unbroken curve. The sine function is continuous for all real numbers.

• Symmetry: The graph is symmetric with respect to the origin. This means that if $(x, y)$ is a point on the graph, then $(-x, -y)$ is also a point on the graph. This is characteristic of an odd function, which satisfies $ f(-x) = -f(x) $.

  $\sin(-x) = -\sin x$

• Intercepts:

  • y-intercept: The graph crosses the y-axis at $x=0$. $ \sin 0 = 0 $. The y-intercept is $(0, 0)$.

  • x-intercepts (Zeros): The graph crosses the x-axis where $y = \sin x = 0$. This occurs at integer multiples of $\pi$. The x-intercepts are at $ x = n\pi $, where $ n $ is any integer ($ \dots, -2\pi, -\pi, 0, \pi, 2\pi, \dots $).

• Maximum and Minimum Values: The maximum value of 1 occurs at $ x = \frac{\pi}{2} + 2n\pi $ for integer $n$. The minimum value of -1 occurs at $ x = \frac{3\pi}{2} + 2n\pi $ for integer $n$.

Graph of Cosine Function ($y = \cos x$)

The graph of the cosine function, $y = \cos x$, is very similar to the graph of the sine function. Like $\sin x$, the value of $\cos x$ for an angle $x$ is given by the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

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Key Points and Shape

Let's evaluate $\cos x$ at the same key angles used for the sine function within one cycle (0 to $2\pi$):

• At $x = 0$ ($0^\circ$): The terminal side is at (1, 0) on the unit circle. The x-coordinate is 1. So, $ \cos 0 = 1 $. This gives the point $(0, 1)$ on the graph. This is a maximum value.

• At $x = \frac{\pi}{2}$ ($90^\circ$): The terminal side is at (0, 1). The x-coordinate is 0. So, $ \cos (\frac{\pi}{2}) = 0 $. This gives the point $(\frac{\pi}{2}, 0)$ on the graph.

• At $x = \pi$ ($180^\circ$): The terminal side is at (-1, 0). The x-coordinate is -1. So, $ \cos \pi = -1 $. This gives the point $(\pi, -1)$ on the graph. This is a minimum value.

• At $x = \frac{3\pi}{2}$ ($270^\circ$): The terminal side is at (0, -1). The x-coordinate is 0. So, $ \cos (\frac{3\pi}{2}) = 0 $. This gives the point $(\frac{3\pi}{2}, 0)$ on the graph.

• At $x = 2\pi$ ($360^\circ$): The terminal side is back at (1, 0). The x-coordinate is 1. So, $ \cos (2\pi) = 1 $. This gives the point $(2\pi, 1)$ on the graph. This completes one cycle, and the values will repeat.

Plotting these key points $(0,1), (\frac{\pi}{2}, 0), (\pi, -1), (\frac{3\pi}{2}, 0), (2\pi, 1)$ and connecting them smoothly results in a wave-like curve, also a sinusoid.

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Graph of $y = \cos x$

The graph of $y = \cos x$ is a continuous wave that extends infinitely in both directions, repeating every $2\pi$.

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Properties of $y = \cos x$ from the Graph

The properties of the cosine function are similar to those of the sine function:

• Domain: The graph exists for all real values of $x$. Domain = $ \mathbb{R} $.

• Range: The minimum value is -1 and the maximum value is 1. Range = $ [-1, 1] $.

  $-1 \leq \cos x \leq 1$

• Periodicity: The graph repeats its complete cycle every $2\pi$ units. The fundamental period is $ 2\pi $.

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

• Continuity: The graph is a smooth, unbroken curve. The cosine function is continuous for all real numbers.

• Symmetry: The graph is symmetric with respect to the y-axis. This means that if $(x, y)$ is a point on the graph, then $(-x, y)$ is also a point on the graph. This is characteristic of an even function, which satisfies $ f(-x) = f(x) $.

  $\cos(-x) = \cos x$

• Intercepts:

  • y-intercept: The graph crosses the y-axis at $x=0$. $ \cos 0 = 1 $. The y-intercept is $(0, 1)$.

  • x-intercepts (Zeros): The graph crosses the x-axis where $y = \cos x = 0$. This occurs at odd multiples of $\pi/2$. The x-intercepts are at $ x = \frac{\pi}{2} + n\pi $, where $ n $ is any integer ($ \dots, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \dots $).

• Maximum and Minimum Values: The maximum value of 1 occurs at $ x = 2n\pi $ for integer $n$. The minimum value of -1 occurs at $ x = \pi + 2n\pi = (2n+1)\pi $ for integer $n$.

• Relationship to Sine Graph: The cosine graph is simply the sine graph shifted horizontally to the left by $\frac{\pi}{2}$ units. This is expressed by the identity $ \cos x = \sin(x + \frac{\pi}{2}) $.

Graph of Tangent Function ($y = \tan x$)

The tangent function, $y = \tan x$, defined as $ \tan x = \frac{\sin x}{\cos x} $, has a graph that is fundamentally different in shape compared to the sine and cosine graphs. This is because the tangent function is not defined for all real numbers, unlike sine and cosine.

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Key Points and Shape

The key features determining the shape of the tangent graph are its undefined points (vertical asymptotes) and its zeros (x-intercepts).

• Vertical Asymptotes: The tangent function $ \tan x = \frac{\sin x}{\cos x} $ is undefined when the denominator $ \cos x = 0 $. As seen from the graph of $y = \cos x$, this occurs at $ x = \frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2}, -\frac{3\pi}{2}, \dots $, which are all angles of the form $ x = \frac{\pi}{2} + n\pi $, where $n$ is any integer. At these values of $x$, the graph of $y = \tan x$ has vertical asymptotes – lines that the graph approaches but never touches.

• Zeros (x-intercepts): The tangent function $ \tan x = \frac{\sin x}{\cos x} $ is zero when the numerator $ \sin x = 0 $ (provided $\cos x \ne 0$ at the same time). As seen from the graph of $y = \sin x$, this occurs at $ x = 0, \pi, 2\pi, -\pi, -2\pi, \dots $, which are all angles of the form $ x = n\pi $, where $n$ is any integer. These are the points where the graph crosses the x-axis.

• Behaviour between Asymptotes: Let's look at the interval $ (-\frac{\pi}{2}, \frac{\pi}{2}) $, which contains the origin. This interval is bounded by two consecutive vertical asymptotes, $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$.

  • At $x = 0$, $ \tan 0 = 0 $. The graph passes through $(0, 0)$.

  • As $x$ increases from $0$ towards $\frac{\pi}{2}$, $\sin x$ increases towards 1, and $\cos x$ decreases towards 0 (remaining positive). The ratio $y/x$ becomes very large positive. So, as $ x \to \frac{\pi}{2}^- $, $ \tan x \to +\infty $.

  • As $x$ decreases from $0$ towards $-\frac{\pi}{2}$, $\sin x$ decreases towards -1 (remaining negative), and $\cos x$ increases towards 0 (remaining positive). The ratio $y/x$ becomes very large negative. So, as $ x \to -\frac{\pi}{2}^+ $, $ \tan x \to -\infty $.

  Plotting these values shows a curve that rises steeply from $-\infty$ near $x = -\pi/2$ to $+\infty$ near $x = \pi/2$, passing through $(0,0)$.

This shape between two consecutive asymptotes is one branch of the tangent graph. Since the function is periodic with period $\pi$, this branch shape repeats in every interval of length $\pi$ between consecutive asymptotes.

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Graph of $y = \tan x$

The graph consists of an infinite number of identical, disconnected branches, separated by vertical asymptotes.

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Properties of $y = \tan x$ from the Graph

From the graph and definition, we can summarise the properties of the tangent function:

• Domain: The function is undefined where $\cos x = 0$. Domain = $ \mathbb{R} - \left\{ x \mid x = \frac{\pi}{2} + n\pi, n \in \mathbb{Z} \right\} $.

• Range: The y-values cover all real numbers along each branch. Range = $ \mathbb{R} $.

• Periodicity: The pattern repeats every $\pi$ units. The fundamental period is $ \pi $.

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

• Continuity: The function is discontinuous at the vertical asymptotes. It is continuous on each open interval between consecutive asymptotes, i.e., on $ (-\frac{\pi}{2} + n\pi, \frac{\pi}{2} + n\pi) $ for any integer $n$.

• Symmetry: The graph is symmetric with respect to the origin. This indicates an odd function.

  $\tan(-x) = -\tan x$

• Vertical Asymptotes: Occur at $ x = \frac{\pi}{2} + n\pi $, where $ n \in \mathbb{Z} $.

• x-intercepts (Zeros): Occur at $ x = n\pi $, where $ n \in \mathbb{Z} $.

• y-intercept: Occurs at $x=0$. $ \tan 0 = 0 $. The y-intercept is $(0, 0)$.

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Note for Competitive Exams

Graphs of Reciprocal Trigonometric Functions ($y = \text{cosec} \, x$, $y = \sec x$, $y = \cot x$)

The graphs of the reciprocal trigonometric functions – cosecant, secant, and cotangent – are derived from the graphs of their corresponding reciprocal functions: sine, cosine, and tangent. Understanding the relationship $ \text{cosec} \, x = \frac{1}{\sin x} $, $ \sec x = \frac{1}{\cos x} $, and $ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} $ is key to sketching and interpreting their graphs.

Since these functions involve reciprocals, they will have vertical asymptotes wherever their reciprocal function is zero.

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1. Graph of Cosecant Function ($y = \text{cosec} \, x$)

The cosecant function is the reciprocal of the sine function: $ \text{cosec} \, x = \frac{1}{\sin x} $. Its graph can be sketched by considering the values of $\sin x$ and taking their reciprocals.

Relation to Sine Graph:

• Wherever $ \sin x = 0 $, $ \text{cosec} \, x $ is undefined. These points are vertical asymptotes.

• Wherever $ \sin x = 1 $ (maximum), $ \text{cosec} \, x = \frac{1}{1} = 1 $ (minimum value of the positive branches).

• Wherever $ \sin x = -1 $ (minimum), $ \text{cosec} \, x = \frac{1}{-1} = -1 $ (maximum value of the negative branches).

• As $ \sin x $ approaches 0 (from positive values), $ \text{cosec} \, x $ approaches $+\infty$.

• As $ \sin x $ approaches 0 (from negative values), $ \text{cosec} \, x $ approaches $-\infty$.

• The range of $ \sin x $ is $ [-1, 1] $. The range of $ \text{cosec} \, x $ is the set of reciprocals of values in $ [-1, 0) \cup (0, 1] $, which is $ (-\infty, -1] \cup [1, \infty) $.

Graph of $y = \text{cosec} \, x$:

The graph consists of repeating U-shaped branches, opening upwards where $\sin x$ is positive and downwards where $\sin x$ is negative, separated by vertical asymptotes.

Properties of $y = \text{cosec} \, x$:

• Domain: All real numbers except where $ \sin x = 0 $. Domain = $ \mathbb{R} - \{ x \mid x = n\pi, n \in \mathbb{Z} \} $.

• Range: $ (-\infty, -1] \cup [1, \infty) $.

• Periodicity: Since the period of $\sin x$ is $2\pi$, the period of $ \text{cosec} \, x $ is also $ 2\pi $.

  $\text{cosec} \, (x + 2\pi) = \text{cosec} \, x$

• Vertical Asymptotes: Occur at $ x = n\pi $, where $ n \in \mathbb{Z} $.

• Symmetry: The graph is symmetric with respect to the origin (like $\sin x$), making it an odd function ($ \text{cosec}(-x) = -\text{cosec} \, x $).

• Intercepts: No y-intercept (undefined at $x=0$). No x-intercepts (since $|\text{cosec} \, x| \geq 1$).

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2. Graph of Secant Function ($y = \sec x$)

The secant function is the reciprocal of the cosine function: $ \sec x = \frac{1}{\cos x} $. Its graph is closely related to the graph of $\cos x$.

Relation to Cosine Graph:

• Wherever $ \cos x = 0 $, $ \sec x $ is undefined. These points are vertical asymptotes.

• Wherever $ \cos x = 1 $ (maximum), $ \sec x = \frac{1}{1} = 1 $ (minimum value of the positive branches).

• Wherever $ \cos x = -1 $ (minimum), $ \sec x = \frac{1}{-1} = -1 $ (maximum value of the negative branches).

• As $ \cos x $ approaches 0 (from positive values), $ \sec x $ approaches $+\infty$.

• As $ \cos x $ approaches 0 (from negative values), $ \sec x $ approaches $-\infty$.

• The range of $ \cos x $ is $ [-1, 1] $. The range of $ \sec x $ is $ (-\infty, -1] \cup [1, \infty) $.

Graph of $y = \sec x$:

The graph consists of repeating U-shaped branches, opening upwards where $\cos x$ is positive and downwards where $\cos x$ is negative, separated by vertical asymptotes. Notice it looks like the cosecant graph shifted horizontally.

Properties of $y = \sec x$:

• Domain: All real numbers except where $ \cos x = 0 $. Domain = $ \mathbb{R} - \left\{ x \mid x = \frac{\pi}{2} + n\pi, n \in \mathbb{Z} \right\} $.

• Range: $ (-\infty, -1] \cup [1, \infty) $.

• Periodicity: Since the period of $\cos x$ is $2\pi$, the period of $ \sec x $ is also $ 2\pi $.

  $\sec (x + 2\pi) = \sec x$

• Vertical Asymptotes: Occur at $ x = \frac{\pi}{2} + n\pi $, where $ n \in \mathbb{Z} $.

• Symmetry: The graph is symmetric with respect to the y-axis (like $\cos x$), making it an even function ($ \sec(-x) = \sec x $).

• Intercepts: y-intercept is $(0, 1)$. No x-intercepts (since $|\sec x| \geq 1$).

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3. Graph of Cotangent Function ($y = \cot x$)

The cotangent function is the reciprocal of the tangent function, $ \cot x = \frac{1}{\tan x} $, and also $ \cot x = \frac{\cos x}{\sin x} $. Its graph has similarities to the tangent graph but is also distinct.

Relation to Sine and Cosine Graphs:

• Vertical Asymptotes: Occur where $ \sin x = 0 $ (the denominator in $\frac{\cos x}{\sin x}$), i.e., at $x = n\pi$ ($..., -\pi, 0, \pi, 2\pi, ...$). These are the same locations as the vertical asymptotes for $\text{cosec} \, x$ and the zeros for $\sin x$.

• Zeros (x-intercepts): Occur where $ \cos x = 0 $ (the numerator in $\frac{\cos x}{\sin x}$), provided $\sin x \ne 0$ at those points. This occurs at $x = \frac{\pi}{2} + n\pi$ ($..., -\pi/2, \pi/2, 3\pi/2, ...$). These are the same locations as the vertical asymptotes for $\tan x$ and $\sec x$, and the zeros for $\cos x$.

• Shape: The graph consists of repeating branches between vertical asymptotes. In the interval $ (0, \pi) $ (bounded by asymptotes $x=0$ and $x=\pi$):

  • As $ x \to 0^+ $, $ \sin x \to 0^+ $ and $ \cos x \to 1 $, so $ \cot x = \frac{\cos x}{\sin x} \to +\infty $. The graph starts high near the y-axis.

  • At $x = \pi/2$, $ \cot(\pi/2) = \frac{\cos(\pi/2)}{\sin(\pi/2)} = \frac{0}{1} = 0 $. The graph crosses the x-axis at $(\pi/2, 0)$.

  • As $ x \to \pi^- $, $ \sin x \to 0^+ $ and $ \cos x \to -1 $, so $ \cot x = \frac{\cos x}{\sin x} \to -\infty $. The graph goes down towards $-\infty$ as it approaches $x=\pi$.

  The graph is always decreasing between consecutive asymptotes.

Graph of $y = \cot x$:

The graph consists of an infinite number of identical, disconnected branches, each decreasing from $+\infty$ to $-\infty$ between consecutive vertical asymptotes.

Properties of $y = \cot x$:

• Domain: All real numbers except where $ \sin x = 0 $. Domain = $ \mathbb{R} - \{ x \mid x = n\pi, n \in \mathbb{Z} \} $.

• Range: The y-values cover all real numbers along each branch. Range = $ \mathbb{R} $.

• Periodicity: The pattern repeats every $\pi$ units. The fundamental period is $ \pi $.

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

• Continuity: The function is discontinuous at the vertical asymptotes. It is continuous on each open interval between consecutive asymptotes, i.e., on $ (n\pi, (n+1)\pi) $ for any integer $n$.

• Symmetry: The graph is symmetric with respect to the origin (like $\tan x$), making it an odd function ($ \cot(-x) = -\cot x $).

• Vertical Asymptotes: Occur at $ x = n\pi $, where $ n \in \mathbb{Z} $.

• x-intercepts (Zeros): Occur at $ x = \frac{\pi}{2} + n\pi $, where $ n \in \mathbb{Z} $.

• y-intercept: No y-intercept (undefined at $x=0$).

Analyzing Graphs (Amplitude, Period, Phase Shift, Vertical Shift)

Understanding the basic graphs of trigonometric functions allows us to identify their key characteristics. These characteristics are then used to analyse and sketch the graphs of transformed trigonometric functions, such as $y = A \sin(B(x - C)) + D$, where A, B, C, and D are constants that alter the basic shape and position.

Although this section focuses on the basic graphs ($y=\sin x, y=\cos x, \dots$), we can observe the fundamental values related to these concepts directly from their graphs.

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1. Amplitude

Concept: The amplitude of a periodic function measures how much the function's value varies from its midline. For sine and cosine functions, it is defined as half the distance between the maximum and minimum values.

Amplitude = $\frac{\text{Maximum Value} - \text{Minimum Value}}{2}$.

The amplitude represents the "height" of the wave from its central resting position (midline).

Observation from Basic Graphs:

• For $y = \sin x$ and $y = \cos x$:

  • Maximum Value = 1

  • Minimum Value = -1

  • Amplitude = $ \frac{1 - (-1)}{2} = \frac{2}{2} = 1 $.

  The basic sine and cosine functions have an amplitude of 1.

• For $y = \tan x, y = \cot x, y = \sec x, y = \text{cosec} \, x$:

  These functions have a range that extends to $ \pm\infty $. They do not have a finite maximum or minimum value. Therefore, the concept of amplitude as defined for sine and cosine is generally not applied to these functions.

Effect of Transformation (Implicit): In a transformed function $y = A \sin(Bx)$, the amplitude becomes $|A|$. It stretches or compresses the graph vertically. For $y = A \cos(Bx)$, the amplitude is also $|A|$.

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2. Period

Concept: The period of a periodic function is the smallest positive horizontal distance over which the graph of the function repeats itself. It is the length of one complete cycle.

Observation from Basic Graphs:

• For $y = \sin x, y = \cos x$:

  One full wave cycle is completed over an interval of $2\pi$. The graph from $0$ to $2\pi$ is identical to the graph from $2\pi$ to $4\pi$, and so on. The fundamental period is $ 2\pi $.

• For $y = \tan x, y = \cot x$:

  One complete branch shape repeats over an interval of $\pi$. For tangent, the interval $(-\pi/2, \pi/2)$ covers one period. For cotangent, the interval $(0, \pi)$ covers one period. The fundamental period is $ \pi $.

• For $y = \sec x, y = \text{cosec} \, x$:

  Although their branches look different, the entire pattern of branches repeats every $2\pi$. Their fundamental period is $ 2\pi $ (same as their reciprocal functions).

Function	Fundamental Period
$\sin x, \cos x, \text{cosec} \, x, \sec x$	$2\pi$
$\tan x, \cot x$	$\pi$

Effect of Transformation (Implicit): In a transformed function $y = f(Bx)$, where $f$ is a trigonometric function, the period of the new function is the period of $f(x)$ divided by $|B|$. For example, the period of $y = \sin(Bx)$ is $ \frac{2\pi}{|B|} $, and the period of $y = \tan(Bx)$ is $ \frac{\pi}{|B|} $.

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3. Phase Shift (Horizontal Shift)

Concept: Phase shift refers to a horizontal translation (shift left or right) of the graph of a periodic function relative to its usual position. It indicates where the cycle "starts".

Observation from Basic Graphs:

• The graph of $y = \cos x$ has the same shape as $y = \sin x$ but is shifted horizontally to the left by $\frac{\pi}{2}$ units. We can see this from the identity $ \cos x = \sin(x + \frac{\pi}{2}) $. The phase shift of the cosine graph relative to the sine graph is $ -\frac{\pi}{2} $ (or $\frac{\pi}{2}$ to the left).

• The graph of $y = \cot x$ is similar to $y = \tan x$ but is shifted. Note the relationship $ \cot x = \tan(\frac{\pi}{2} - x) = -\tan(x - \frac{\pi}{2}) $. This indicates a shift and a reflection.

Effect of Transformation (Implicit): In a transformed function of the form $y = f(x - C)$ or $y = f(Bx - C) = f(B(x - C/B))$, the graph is shifted horizontally. The term $(x - C)$ indicates a shift of $C$ units to the right. The term $(x + C)$ indicates a shift of $C$ units to the left. In the form $f(B(x - C/B))$, the phase shift is $C/B$.

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4. Vertical Shift

Concept: A vertical shift is an upward or downward translation of the entire graph of the function. It changes the position of the midline of the graph.

Observation from Basic Graphs:

• The basic graphs of $y = \sin x$ and $y = \cos x$ oscillate symmetrically around the x-axis ($y=0$). The x-axis acts as their midline.

• The basic graphs of $y = \tan x$ and $y = \cot x$ are centered vertically on the x-axis, though they extend infinitely in both directions.

Effect of Transformation (Implicit): In a transformed function $y = f(x) + D$, the entire graph is shifted vertically by $D$ units. If $D > 0$, the shift is upwards; if $D < 0$, the shift is downwards. The midline for $y = A \sin(B(x-C)) + D$ or $y = A \cos(B(x-C)) + D$ is the horizontal line $y = D$.

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Summary of Key Properties of Basic Graphs

Function	Domain	Range	Period	Key Asymptotes/Zeros (within one period)
$y = \sin x$	$\mathbb{R}$	$[-1, 1]$	$2\pi$	Zeros at $0, \pi, 2\pi$
$y = \cos x$	$\mathbb{R}$	$[-1, 1]$	$2\pi$	Zeros at $\pi/2, 3\pi/2$
$y = \tan x$	$\mathbb{R} - \{ \frac{\pi}{2} + n\pi \}$	$\mathbb{R}$	$\pi$	Vertical Asymptotes at $\pi/2, 3\pi/2, \dots$ (or $-\pi/2, \pi/2, \dots$); Zeros at $0, \pi, 2\pi, \dots$
$y = \text{cosec} \, x$	$\mathbb{R} - \{ n\pi \}$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	Vertical Asymptotes at $0, \pi, 2\pi, \dots$
$y = \sec x$	$\mathbb{R} - \{ \frac{\pi}{2} + n\pi \}$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	Vertical Asymptotes at $\pi/2, 3\pi/2, \dots$
$y = \cot x$	$\mathbb{R} - \{ n\pi \}$	$\mathbb{R}$	$\pi$	Vertical Asymptotes at $0, \pi, 2\pi, \dots$; Zeros at $\pi/2, 3\pi/2, \dots$

(Note: $n \in \mathbb{Z}$ in the domain/asymptote descriptions).

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Note for Competitive Exams