Domains, Ranges and Principal Values

General Solutions of Trigonometric Equations: Your JEE Main Key

Hey there, future IITians! Trigonometric equations might seem daunting at first, but mastering them is crucial for acing the JEE Main. Understanding general solutions allows you to tackle a wide range of problems involving angles and their relationships. In this lesson, we'll unlock the secrets of general solutions for basic trigonometric equations, equipping you with the tools to conquer those challenging problems.

Understanding General Solutions: Beyond the Principal Value

When you solve a trigonometric equation like $\sin x = \frac{1}{2}$, you might immediately think of $x = \frac{\pi}{6}$ (30 degrees). This is the principal solution – the solution that lies within the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ for sine, $[0, \pi]$ for cosine, and $(-\frac{\pi}{2}, \frac{\pi}{2})$ for tangent. However, there are infinitely many other angles that satisfy the same equation. These are the general solutions. The general solution expresses all possible solutions using the periodicity of trigonometric functions.

1. General Solution of $\sin x = \sin \alpha$

Imagine the unit circle. If $\sin x = \sin \alpha$, both $x$ and $\alpha$ have the same vertical coordinate. This happens not only at $\alpha$ but also at $\pi - \alpha$. Due to the periodic nature of the sine function, we can add multiples of $2\pi$ to these angles and still get the same sine value. This leads us to the general solution:

Where $n$ is any integer ($n \in \mathbb{Z}$).

Explanation:

• When $n$ is even, $x = n\pi + \alpha$. This represents angles that are $n$ full rotations ($2\pi$) plus $\alpha$.
• When $n$ is odd, $x = n\pi - \alpha$. This represents angles that are an odd number of $\pi$ away, minus $\alpha$, effectively capturing the $\pi - \alpha$ solution and its periodic equivalents.

Example: Solve $\sin x = \sin \frac{\pi}{4}$. Using the formula, $x = n\pi + (-1)^n\frac{\pi}{4}$, where $n \in \mathbb{Z}$.

2. General Solution of $\cos x = \cos \alpha$

Similarly, for $\cos x = \cos \alpha$, both $x$ and $\alpha$ have the same horizontal coordinate on the unit circle. This occurs at $\alpha$ and $-\alpha$. Adding multiples of $2\pi$ gives us the general solution:

Where $n$ is any integer ($n \in \mathbb{Z}$).

Explanation: This formula captures both $\alpha$ and $-\alpha$ (which is equivalent to $2\pi - \alpha$) and all their periodic equivalents (adding multiples of $2\pi$).

Example: Solve $\cos x = \cos \frac{\pi}{3}$. Using the formula, $x = 2n\pi \pm \frac{\pi}{3}$, where $n \in \mathbb{Z}$.

3. General Solution of $\tan x = \tan \alpha$

For $\tan x = \tan \alpha$, both $x$ and $\alpha$ have the same ratio of sine to cosine. The tangent function has a period of $\pi$, so we simply add multiples of $\pi$ to $\alpha$ to obtain the general solution:

Where $n$ is any integer ($n \in \mathbb{Z}$).

Explanation: The tangent function repeats every $\pi$ radians, so adding $n\pi$ accounts for all possible solutions.

Example: Solve $\tan x = \tan \frac{\pi}{6}$. Using the formula, $x = n\pi + \frac{\pi}{6}$, where $n \in \mathbb{Z}$.

4. Special Cases: $\sin x = 0$ and $\cos x = 0$

These are frequently encountered scenarios and should be memorized:

Where $n$ is any integer ($n \in \mathbb{Z}$). Sine is zero at integer multiples of $\pi$ (0, $\pi$, $2\pi$, $-\pi$, etc.).

Where $n$ is any integer ($n \in \mathbb{Z}$). Cosine is zero at odd multiples of $\frac{\pi}{2}$ ($\frac{\pi}{2}$, $\frac{3\pi}{2}$, $-\frac{\pi}{2}$, etc.).

Principal and General Solutions: A Clear Distinction

Remember, the principal solution is a specific solution within a defined interval, while the general solution represents all possible solutions. When solving trigonometric equations, always find the general solution unless specifically asked for the principal solution.

Common Mistakes to Avoid

JEE Main Tricks and Strategies

JEE Main problems often involve manipulating trigonometric equations before applying the general solution formulas. Here's a common strategy:

1. Simplify the equation: Use trigonometric identities to simplify the given equation into one of the standard forms ($\sin x = \sin \alpha$, $\cos x = \cos \alpha$, $\tan x = \tan \alpha$).
2. Find the principal solution: Determine the value of $\alpha$ that satisfies the simplified equation.
3. Apply the general solution formula: Substitute the value of $\alpha$ into the appropriate formula to obtain the general solution.

Mastering general solutions of trigonometric equations is essential for success in JEE Main. By understanding the concepts, memorizing the formulas, and practicing regularly, you'll be well-equipped to tackle any trigonometric equation problem that comes your way. Keep practicing, and best of luck!