Problem 40 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\sum_{i=1}^{2 n} \sin ^{-1} x_{i}=n \pi\), then \(\sum_{i=1}^{2 n} x_{i}\) is equal to (A) \(n\) (B) \(2 n\) (C) \(\frac{n(n+1)}{2}\) (D) none of these

Step-by-Step Solution

Answer

The sum \( \sum_{i=1}^{2n} x_i \) in this expression is \(0\).

Step 1: Understanding the Problem

We are given that \( \sum_{i=1}^{2n} \sin^{-1} x_i = n \pi \). We need to find the value of \( \sum_{i=1}^{2n} x_i \). The sum of inverse sine functions for various values \( x_i \) is equal to \( n \pi \).

Step 2: Analyzing the Expression

The range of \( \sin^{-1}(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). A single \( \sin^{-1}(x_i) \) can take values from this range. For \( \sum_{i=1}^{2n} \sin^{-1} x_i = n \pi \), each \( \sin^{-1} x_i \) must take specific values such that their sum results in \( n \pi \).

Step 3: Finding Required Values

To achieve the sum \( n \pi \) with \( 2n \) terms, we may consider that half of the \( x_i \)'s are 1, so that their \( \sin^{-1}(x_i) = \frac{\pi}{2} \), and the other half are -1, so \( \sin^{-1}(x_i) = -\frac{\pi}{2} \). This way, each pair sums up to 0. However, intuitively, that's not possible here because we need them to add up to positive sums.

Step 4: Revising Hypothesis

We consider if all \( x_i \)'s are zero, then \( \sin^{-1}(x_i) \) is zero, but this doesn't fit as the sum becomes zero. If half are zero and half are 1, those that are 1 continue to contribute \( \frac{\pi}{2} \) each.

Step 5: Simplification and Calculation

Suppose all \( x_i \) are identical and equal to 1 and -1, we have: sum of integer values equaling zero making pair wise sum zero which means \( n \pi \) pairs crucial among remaining are 0.

Step 6: Sum of x-values

With the intuitive correction where sum pi occurs which implies x-values to 0 might infer change, we validate if \( x_i = \cos(0), \sin^{-1}(x_i)=\pi/2 \) critiquing total we simplify cases around coefficient consistency.

Step 7: Conclusion

As the adjustability within arange zero target matches \( x_i = 0 \) hence sum is default zero which also remain beyond range alignment into variable coefficient measures ensuring consistent unity assessment given potential structural equilibrium.

Key Concepts

Inverse Sine Function

The inverse sine function, commonly represented as \( \sin^{-1}(x) \), is a key tool in trigonometry. This function allows you to determine the angle whose sine is \( x \). When you see \( \sin^{-1}(x) \), it means "what angle has a sine value of \( x \)?"
This function has a specific range: from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This range ensures that each sine value corresponds to one unique angle, which is necessary for the function to be "invertible."
Within this range, if a value is given for \( x \), you can find a corresponding angle \( \theta \) such that \( \sin(\theta) = x \). For instance:

• \( \sin^{-1}(0) = 0 \)
• \( \sin^{-1}(1) = \frac{\pi}{2} \)
• \( \sin^{-1}(-1) = -\frac{\pi}{2} \)

In the exercise, using the inverse sine function helped reach the conclusion about the sum of angles being equal to \( n \pi \), setting a challenging requirement for the sum of input values \( x_i \).

Sum of Angles

The concept of the sum of angles is fundamental in solving trigonometric equations, particularly involving multiple terms. When dealing with \( \sum_{i=1}^{2n} \sin^{-1} x_i = n \pi \), the focus is on understanding how multiple angles relate to a larger angle (in this case, \( n \pi \)).
This concept often involves partitioning angles in a strategic manner to achieve a desired total sum. In our problem, achieving \( n \pi \) with the sum of \( 2n \) instances of \( \sin^{-1} x_i \) requires a balance in values attributed to each angle:

• For instance, with each \( \sin^{-1}(1) = \frac{\pi}{2} \), and \( \sin^{-1}(-1) = -\frac{\pi}{2} \), each pair can be calibrated to zero, implying polar opposite angles.
• If all \( x_i \) values were identical, say \( 1 \) or \(-1\), then while the \( \sin^{-1} x_i \) values were consistent, achieving \( n \pi \) wouldn't be feasible without carefully choosing \( x_i \).

Understanding the sum of angles helps in structuring inputs and aligning them with the expected total, often requiring insight into equal partitions or complementary angle pairs.

Angle Sum and Difference Identities

The angle sum and difference identities play a crucial role in trigonometry by providing relationships between the sines and cosines of sums and differences of angles.
These identities are especially useful when dealing with the sum or difference of angles to solve problems involving inverse trigonometric functions. Although they were not directly applied to find \( \sum_{i=1}^{2n} x_i \), understanding them aids in breaking down more complex trigonometric expressions.
The basic forms of these identities are:

• Sum: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• Difference: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
• Sum: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• Difference: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

Understanding these can simplify the handling of problems involving trigonometric sums and differences, whether in theoretical applications or practical scenarios.