Problem 22 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the number of solutions of (i) \(2^{\sin x}=\sin x, \forall x \in[0,4 \pi]\) (ii) \(2^{\sin x}=|\sin x|, \forall x \in[0,4 \pi]\) (iii) \(2^{\cos x}=|\sin x|, \forall x \in[0,8 \pi]\) (iv) \(2^{\cos x}=|\cos x|, \forall x \in[0,10 \pi]\) (v) \(2^{\cos x}=\cos x, \forall x \in[0,20 \pi]\) (vi) \(x \sin x-1=0, \forall x \in[0,2 \pi]\) (vii) \(x \sin ^{2} x-1=0, \forall x \in[0,2 \pi]\) (viii) \(\sin x=2^{x}+2^{x}, \forall x \in[0,2 \pi]\) (ix) \(\cos x=2^{x}-2^{x}, \forall x \in[0,2 \pi]\) (x) \(x^{2} \cos x-2=0, \forall x \in[0,2 \pi]\)

Step-by-Step Solution

Answer

The number of solutions are hard to define without knowing the precise plots of the functions. It's recommended to use a graphing tool to visualize it. Thus, the short answer will depend on the number of intersections found in each graph, within the given range of values.

Step 1: Graph the functions

The first step is to graph the functions for each exercise. Using a plotter, draw two lines on the same graph, one for each side of the equation. The intervals are given, so that's the range within which to plot.

Step 2: Find Intersections

After plotting the two functions on the same graph, look for points where the functions intersect. These intersections represent the solutions of the equations.

Step 3: Count Intersections

Once the intersections are identified, count them. The number of these intersections would be the number of solutions for that particular equation in the given interval.

Key Concepts

Graphical Solutions of Equations

Graphical solutions offer a visual approach to understanding and solving equations. By plotting both sides of an equation on a graph, you can easily spot where the two functions intersect.
Each intersection represents a solution to the equation, meaning it's a point where both functions share the same value for the given input of x. People often use this method because:

• It's intuitive and provides a clear visual of solutions.
• It circumvents complex algebraic manipulations, especially with transcendental equations.

When approaching graphical solutions:

• Ensure the graphs cover the entire given interval to miss no intersections.
• Use a precise plotting tool for capturing even subtle intersecting points.

Graphical methods are particularly effective when traditional algebraic techniques are cumbersome.

Interval Analysis

Interval analysis is a crucial technique in finding solutions to equations, especially when dealing with trigonometric and transcendental equations. It involves analyzing the behavior of functions within a specified range or interval to find solutions where certain conditions, like equality, are met.
This approach is used because functions can behave differently over different intervals, especially trigonometric functions that are periodic. There are some key points to remember about interval analysis:

• Breaking down a problem into smaller intervals can simplify the analysis.
• Different intervals might yield different numbers of solutions.
• Identifying symmetry or periodicity within an interval can reduce the workload.

Applying interval analysis effectively requires a solid understanding of the function's behavior throughout the interval, ensuring that all potential solutions are considered.

Solutions of Transcendental Equations

Transcendental equations involve non-algebraic functions, such as exponential, logarithmic, or trigonometric functions. These equations often don't have straightforward algebraic solutions, making numerical or graphical methods more suitable.
Solutions of transcendental equations can be challenging due to their complex nature. Often, multiple techniques are combined to find solutions, such as:

• Graphical methods to visually find intersections.
• Iterative numerical methods, like the Newton-Raphson method, to approximate solutions.
• Interval narrowing to find more accurate solution estimates within a range.

When working with transcendental equations, expect to verify solutions using different approaches to ensure accuracy, as these equations may have multiple solutions or none at all, depending on the interval analyzed.