Problem 12 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(x+y=\frac{\pi}{4}\) and \(\tan x+\tan y=1\)

Step-by-Step Solution

Answer

So, the solutions to the equations are \(x = \frac{\pi}{4} - \frac{1}{2}\) and \(y = \frac{1}{2} - \frac{\pi}{4}\)

Step 1: Add and Subtract the Equations

Start by adding the two equations together and then subtracting the second equation from the first one: \(2x = \frac{\pi}{2} - 1\), \(2y = 1 - \frac{\pi}{2}\)

Step 2: Simplify the equations

Next, simplify both equations to solve for \(x\) and \(y\). Divide each equation by 2: \(x = \frac{\frac{\pi}{2} - 1}{2}\), \(y = \frac{1 - \frac{\pi}{2}}{2} \). This way, we can find the values of \(x\) and \(y\) relative to \(\frac{\pi}{4}\)

Step 3: Calculate the Values

After simplifying and solving, we have: \(x = \frac{\pi}{4} - \frac{1}{2}\), \(y = \frac{1}{2} - \frac{\pi}{4}\)

Key Concepts

System of Trigonometric Equations

Understanding the system of trigonometric equations is crucial for students tackling advanced mathematics problems. Such a system includes two or more equations that incorporate trigonometric functions and require simultaneous solutions. In our example,

We encounter a system with two equations:

• The first is a linear equation in terms of angles, ewline \(x + y = \frac{\pi}{4}\) ewline , which imposes a direct relationship between two variables.
• The second is a trigonometric equation that depends on the tangent function, ewline \(\tan x + \tan y = 1\) ewline .

To approach this system, one common method is to manipulate the equations—often by adding, subtracting, or substituting—to isolate one variable at a time. Through this strategy, the complexity of the equations reduces, making it easier to solve for the unknown angles. It can be beneficial to remember that these systems often have multiple solutions due to the periodic nature of trigonometric functions.

Solving Trigonometric Equations

Solving trigonometric equations often involves algebraic manipulation and a strong understanding of trigonometric identities and properties. The step-by-step solution provided is a classic algebraic approach to solving the system:

1. Adding and subtracting equations neutralizes one of the variables, allowing us to solve for the other.
2. Simplifying the equations further by algebraic means—here, dividing by 2 makes the equations more manageable.
3. Finally, calculating the values of the variables involves direct computation, applying the simplifications already performed.

The trickier aspect here is to understand the tangent function's role and to consider its properties—that it is periodic and undefined at certain points (multiples of \(\frac{\pi}{2}\)). These features may lead to extra solutions or to restrictions in the possible range of solutions.

Tangent Function Properties

The tangent function, represented as \(\tan\), has unique properties that can strongly influence the solutions to trigonometric equations involving it. Some key characteristics include:

• Periodicity: The tangent function has a period of \(\pi\), meaning it repeats its values every \(\pi\) radians.
• Symmetry: It is an odd function, implying that \(\tan(-x) = -\tan(x)\).
• Undefined points: Tangent is undefined where the cosine function is zero, namely at odd multiples of \(\frac{\pi}{2}\).
• Infinitely many solutions: Since tangent is periodic, an equation involving \(\tan\) can have an infinite number of solutions unless the domain is restricted.

When solving equations like \(\tan x + \tan y = 1\), a keen sense of these properties helps identify solutions and avoid common mistakes. For instance, considering the periodicity, we should be mindful that there may be more than one pair of angles \((x, y)\) that satisfy the given system.