Problem 14 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following trigonometric equations: Find the number of roots of \(\cos x-x+\frac{1}{2}=0\) lies in \(\left(0, \frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

The number of roots for the given equation \(\cos x - x+\frac{1}{2}=0\) in the interval \(0, \frac{\pi}{2}\) is 1.

Step 1: Rewrite the equation

The problem is to solve \(\cos x -x +\frac{1}{2}=0\). It can be rewritten as \(\cos x= x-\frac{1}{2}\).

Step 2: Locate roots within the interval

Draw the graph of the functions \(y = \cos x\) and \(y = x-\frac{1}{2}\) and find their intersection points in the domain \(0, \frac{\pi}{2}\)

Step 3: Check Intersection Points

By analyzing the graphs, it can be seen that there is only one intersection point. Therefore, there is only one root for the given equation in the interval \(0, \frac{\pi}{2}\).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in the study of triangles and modeling periodic phenomena. The cosine function, denoted as \(\cos x\), expresses the ratio of the adjacent side to the hypotenuse in a right-angled triangle when given an angle \(x\).

Understanding the cosine function is pivotal, especially when solving trigonometric equations. The function exhibits a wave-like pattern, oscillating between \(1\) and \( -1 \), and is periodic with period \(2\pi\). This periodic nature of cosine means it will repeat its values over every interval of \(2\pi\), making prediction of its behavior in any given interval possible.

In the context of the exercise \(\cos x - x + \frac{1}{2} = 0\), recognizing the behavior of the cosine function helps in estimating possible values of \(x\) that will satisfy the equation. As the cosine value decreases from \(1\) to \(0\) in the first quadrant (\(0 < x < \frac{\pi}{2}\)), our task is to find where it intersects with the linear function \(x - \frac{1}{2}\).

Graphical Method

The graphical method to solving equations involves plotting the functions on a Cartesian plane and visually identifying the points where they intersect. In our example, we construct two graphs: one for \(y = \cos x\) and another for \(y = x - \frac{1}{2}\).

Graphing these functions provides a clear visual of how the more predictable, smooth wave of the cosine function compares to the straight line's consistent rise. Intersection points of the two graphs correspond to shared solutions of the equation, as both \(y\)-values are equal at these \(x\)-coordinates.

The advantage of this method is its visual nature, making it easier to understand. However, accurately finding the exact intersection points can often be challenging without further mathematical tools or technology. Nonetheless, for the purpose of counting the number of roots within an interval, as required by the exercise, a sketch can be sufficient.

Equation Roots

In mathematics, the term 'root' refers to the solution of an equation; it's the value of the variable that makes the equation true. When we look at \(\cos x - x + \frac{1}{2} = 0\), we are essentially seeking the roots of this equation within a specified interval, \(0, \frac{\pi}{2})\).

Determining the number of roots within an interval can often be done by employing the graphical method, as it provides a count of how many times the functions intersect. In the given exercise, after plotting the two equations, we observe there’s a single point where \(y = \cos x\) crosses \(y = x - \frac{1}{2}\) between \(0\) and \(\frac{\pi}{2}\), which means there is exactly one root in the given interval.

It’s crucial to understand that while other roots may exist outside the interval, they are not considered in the context of the exercise. Therefore, special attention is given to the domain specified in the problem when searching for roots.