Problem 20 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\sum_{i=1}^{2 n} \sin ^{-1} x_{i}=n \pi\), then \(\sum_{i=1}^{2 n} x_{j}\) is (a) \(n\) (b) \(2 n\) (c) \(\frac{n(n+1)}{2}\) (d) \(\frac{n(n-1)}{2}\)

Step-by-Step Solution

Answer

Thus, the correct answer is (b) \(2n\). This indicates that the sum of all the \(x_k\) is \(2n\).

Step 1: Understand Inverse Sine Properties

Since the inverse sine \( \sin^{-1}(x_k) \) is defined for x in the interval \(-1 \leq x \leq 1\), the output of \( \sin^{-1}{x_k} \) can range between -\(\pi/2\) to \(\pi/2\). The given that the sum is \(n \pi\), which is a positive value, forces all the \(x_k\) to be 1. Because the inverse sine of 1, \( \sin^{-1}{1} \), gives \(\pi/2\). Therefore \(x_k = 1\) for \(k = 1\) to \(2n\).

Step 2: Sum of \(x_k\)

Now since all the \(x_k\) for \(k = 1\) to \(2n\) are equal to 1, the sum of all these \(x_k\) becomes equal to \(2n\). So, \(\sum_{k=1}^{2n}x_k = 2n\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are fundamental in both pure and applied mathematics. They allow us to find angles based on known ratios. Specifically, the inverse sine function, denoted as \(\sin^{-1}(x)\) or \(\arcsin(x)\), takes a real number input \(x\) usually between \(-1\) and \(1\), and returns an angle from \(-\pi/2\) to \(\pi/2\). This means that when solving problems, one must ensure the appropriate range for the output angles.

In the exercise, it was given that the sum of several inverse sines equals \(n \pi\), implying that each individual part, \(\sin^{-1}(x_i)\), has a range contributing factors of \(\pi/2\) to sum up to \(n \pi\). Hence, for their sum to consistently meet this requirement, each \(x_i\) must be 1 because \(\sin^{-1}(1) = \pi/2\). This critical conclusion allows us to interpret the problem and simplifies the computations significantly.

Sine Properties

The sine function, one of the primary trigonometric functions, varies periodically between -1 and 1. By understanding its behavior, we can leverage these properties to ascertain useful identities and relationships in trigonometry. Inverse sine directly relates to this function as it finds angles whose sine is a given number.

For example, when \(\sin(x) = 1\), the angle is either \(\pi/2\) or \(90°\). This supports our step in the solution where each \(\sin^{-1}(x_i)\) could peak only when \(x_i = 1\) for the whole expression to add up correctly to the required \(n\pi\).

Remembering such identifiers and relationships allows easier problem solving in advanced trigonometric exercises.

Summation in Trigonometry

Summation in trigonometry often deals with adding sequential trigonometric values or their derivatives. This is not only useful in exercises but also in real-life applications like wave functions or signal processing. For the given problem, recognizing the summation formula involves a straightforward technique: recognizing patterns and functions of the elements in sequence.

Once it was established that \(x_j = 1\) for each term, the summation of these constant terms simply becomes \(2n\). This involves basic summation logic where adding \(1\) repeatedly results in counting up all terms, reaching \(2n\). This informs us without needing complex tools or formulas explicitly. Thus, connecting the sum directly to the number of elements and their magnitude quickly leads to the correct conclusion of the exercise.