Problem 28 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(\frac{\cos x}{1+\cos 2 x}=0\)

Step-by-Step Solution

Answer

The solutions to the equation \(\frac{\cos x}{1+\cos 2 x}=0\) are given by \(x=(2n+1)\frac{\pi}{2}\), for all integers \(n\).

Step 1: Analyze the equation

Make an observation that a fraction is equals to zero when its numerator is zero. Hence from \(\frac{\cos x}{1+\cos 2 x}=0\), \(\cos x=0\) because \(\cos x\) is the numerator.

Step 2: Find the roots of the equation

\(\cos x=0\) has solution when \(x=(2n+1)\frac{\pi}{2}\), for all integer values of \(n\). Hence the roots of the equation are \((2n+1)\frac{\pi}{2}\).

Key Concepts

Understanding the Cosine Function

The cosine function is one of the basic trigonometric functions, essential in understanding angles and their relationships in mathematics. Essentially, the cosine of an angle \((x)\) gives the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is usually denoted by \(\cos x\). Understanding this function is crucial in analyzing various trigonometric equations.

• Range: The output or range of the cosine function is from -1 to 1.
• Periodicity: The cosine function is periodic with a period of \(2\pi\), meaning every \(2\pi\) units, it repeats its values.
• Even Function: It is an even function, meaning \(\cos(-x) = \cos(x)\).

Recognizing these properties helps in solving equations, as it allows for anticipation of pattern repetition and how roots of these equations might behave.

Solving Trigonometric Equations

Solving equations involving trigonometric functions can seem complex, but once you break them down step by step, they become manageable. Let's consider the basic method involved when solving trigonometric equations, using our example to illustrate:

• Identify the equation: In the given problem, the equation is \( \frac{\cos x}{1+\cos 2x} = 0 \). Here, we notice it is a fraction set to zero.
• Focus on the numerator: In a fraction, it equals zero when its numerator is zero. So, solve \( \cos x = 0 \).

The goal when solving these equations is to isolate the trigonometric function and use its known properties to find the solutions. Trigonometric identities often come in handy, as they offer forms we can work with or simplify.

Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation, making the equation true. Finding these roots is essentially the goal of solving equations.
In our trigonometric context, after isolating \(\cos x = 0\), the roots are derived using the cosine function's behavior.

• The cosine function equals zero at specific points within its periodic cycle. This occurs at odd multiples of \(\frac{\pi}{2}\), specifically when \(x = (2n+1)\frac{\pi}{2}\).
• Here, \(n\) must be an integer, as it represents the number of cycles.

These roots are crucial as they illustrate exact solutions, offering insights into not just single answers but an infinite set bound by the period of the cosine function. Understanding these concepts equips students with the ability to analyze similar trigonometric scenarios effectively.