Graphs of Trigonometric Functions

Graphs of Trigonometric Functions

Hello JEE aspirants! Understanding the graphs of trigonometric functions is absolutely crucial for acing problems in coordinate geometry, calculus, and even physics. This lesson dives deep into the characteristics, transformations, and important formulas related to these graphs, equipping you with the tools to tackle even the trickiest JEE questions.

1. Basic Trigonometric Graphs

Let's start with the fundamental trigonometric functions. Visualizing these graphs will help you understand their properties and behavior.

a) $\sin x$

The graph of $y = \sin x$ oscillates between -1 and 1. It passes through the origin (0, 0) and completes one full cycle from 0 to $2\pi$. Think of it as the height of a point moving around the unit circle.

• Domain: All real numbers ($-\infty < x < \infty$)
• Range: $[-1, 1]$
• Period: $2\pi$

b) $\cos x$

The graph of $y = \cos x$ also oscillates between -1 and 1, but it starts at (0, 1). It's essentially a $\sin x$ graph shifted to the left by $\frac{\pi}{2}$.

• Domain: All real numbers ($-\infty < x < \infty$)
• Range: $[-1, 1]$
• Period: $2\pi$

c) $\tan x$

The graph of $y = \tan x = \frac{\sin x}{\cos x}$ is quite different. It has vertical asymptotes where $\cos x = 0$, i.e., at $x = \frac{(2n+1)\pi}{2}$, where $n$ is an integer. It repeats every $\pi$ units.

• Domain: All real numbers except $x = \frac{(2n+1)\pi}{2}$, where $n$ is an integer
• Range: All real numbers ($-\infty < y < \infty$)
• Period: $\pi$

d) $\cot x$

The graph of $y = \cot x = \frac{\cos x}{\sin x}$ has vertical asymptotes where $\sin x = 0$, i.e., at $x = n\pi$, where $n$ is an integer. It's the reciprocal of $\tan x$ and also repeats every $\pi$ units.

• Domain: All real numbers except $x = n\pi$, where $n$ is an integer
• Range: All real numbers ($-\infty < y < \infty$)
• Period: $\pi$

e) $\sec x$

The graph of $y = \sec x = \frac{1}{\cos x}$ has vertical asymptotes at the same points as $\tan x$ (where $\cos x = 0$). Its range is $(-\infty, -1] \cup [1, \infty)$.

• Domain: All real numbers except $x = \frac{(2n+1)\pi}{2}$, where $n$ is an integer
• Range: $(-\infty, -1] \cup [1, \infty)$
• Period: $2\pi$

f) $\csc x$

The graph of $y = \csc x = \frac{1}{\sin x}$ has vertical asymptotes at the same points as $\cot x$ (where $\sin x = 0$). Its range is also $(-\infty, -1] \cup [1, \infty)$.

• Domain: All real numbers except $x = n\pi$, where $n$ is an integer
• Range: $(-\infty, -1] \cup [1, \infty)$
• Period: $2\pi$

2. Period, Amplitude, and Phase Shift

These parameters help us understand how trigonometric functions are stretched, compressed, and shifted.

a) Period

The period is the length of one complete cycle of the function. For basic $\sin x$ and $\cos x$, the period is $2\pi$. But what happens when we have $\sin(ax + b)$? This is where the formula comes in handy.

Explanation: The '$a$' value affects the horizontal compression or stretching of the graph. If $|a| > 1$, the graph is compressed, and the period decreases. If $|a| < 1$, the graph is stretched, and the period increases. The absolute value ensures that the period is always positive.

Example: Consider $\sin(2x)$. Here, $a = 2$, so the period is $\frac{2\pi}{2} = \pi$. The graph completes a cycle in $\pi$ units instead of $2\pi$ units.

b) Amplitude

The amplitude is the maximum displacement of the graph from its midline (the horizontal line that runs through the middle of the graph). For basic $\sin x$ and $\cos x$, the amplitude is 1. Now, let's see the impact of $A$ in $A\sin(ax + b)$.

Explanation: '$A$' affects the vertical stretching of the graph. If $|A| > 1$, the graph is stretched vertically. If $|A| < 1$, the graph is compressed vertically. If $A$ is negative, the graph is reflected about the x-axis. The absolute value ensures amplitude is always positive.

Example: For $3\cos x$, the amplitude is $|3| = 3$. The graph oscillates between -3 and 3.

c) Phase Shift

Phase shift refers to the horizontal shift of the graph. In the function $A\sin(ax + b)$, the phase shift is given by $-\frac{b}{a}$.

Explanation: The '$b$' value shifts the graph horizontally. A positive value of $-\frac{b}{a}$ shifts the graph to the left, and a negative value shifts it to the right. This concept is directly related to transformations of functions.

Example: Consider $\sin(x + \frac{\pi}{4})$. The phase shift is $-\frac{\pi}{4}$, meaning the graph is shifted $\frac{\pi}{4}$ units to the left.

3. Domain and Range of Trigonometric Functions

We already covered this above for the basic trigonometric functions. It’s essential to remember these and how transformations affect them.

4. Transformations of Trigonometric Graphs

Transformations include vertical and horizontal stretches/compressions, reflections, and translations. Understanding these is crucial for sketching graphs quickly.

• Vertical Stretch/Compression: $y = Af(x)$ stretches the graph vertically by a factor of $|A|$.
• Horizontal Stretch/Compression: $y = f(ax)$ compresses the graph horizontally by a factor of $|a|$.
• Reflection about x-axis: $y = -f(x)$ reflects the graph about the x-axis.
• Reflection about y-axis: $y = f(-x)$ reflects the graph about the y-axis.
• Vertical Translation: $y = f(x) + k$ shifts the graph vertically by $k$ units.
• Horizontal Translation (Phase Shift): $y = f(x + h)$ shifts the graph horizontally by $h$ units.

Let’s combine $a\sin x + b\cos x$!

Derivation: We can rewrite $a\sin x + b\cos x$ in the form $R\sin(x + \alpha)$, where $R = \sqrt{a^2 + b^2}$ and $\alpha = \tan^{-1}(\frac{b}{a})$.

Then, $a\sin x + b\cos x = \sqrt{a^2 + b^2} \left( \frac{a}{\sqrt{a^2 + b^2}}\sin x + \frac{b}{\sqrt{a^2 + b^2}}\cos x \right)$.

Let $\cos \alpha = \frac{a}{\sqrt{a^2 + b^2}}$ and $\sin \alpha = \frac{b}{\sqrt{a^2 + b^2}}$.

Then, $a\sin x + b\cos x = \sqrt{a^2 + b^2} (\cos \alpha \sin x + \sin \alpha \cos x) = \sqrt{a^2 + b^2} \sin(x + \alpha)$.

Since the range of $\sin(x + \alpha)$ is $[-1, 1]$, the range of $a\sin x + b\cos x$ is $[-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]$.

Example: Consider $3\sin x + 4\cos x$. Here, $a = 3$ and $b = 4$. The range is $[-\sqrt{3^2 + 4^2}, \sqrt{3^2 + 4^2}] = [-5, 5]$.

Remember to practice plotting these graphs and applying these transformations. Good luck with your JEE preparation!