Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\cos ^{-1}\left(\frac{n}{2 \pi}\right)>\frac{2 \pi}{3}\) then the minimum and the maximum values of integer \(n\) are respectively (A) \(-6\) and \(-4\) (B) 4 and 6 (C) \(-6\) and \(-3\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A): -6 and -4.

Step 1: Understand the Given Inequality

We are given \( \cos^{-1}\left(\frac{n}{2\pi}\right) > \frac{2\pi}{3} \).This inequality involves the inverse cosine function, which means we need to consider the range and domain of the cosine function. The range of \( \cos^{-1}(x) \) is \([0, \pi]\).

Step 2: Understand the Range of Cosine

The inequality \( \cos^{-1}\left(\frac{n}{2\pi}\right) > \frac{2\pi}{3} \) implies that \( \frac{n}{2\pi} < \cos\left(\frac{2\pi}{3}\right) \). The value of \( \cos\left(\frac{2\pi}{3}\right) \) is \(-\frac{1}{2}\). Therefore, the inequality becomes:\( \frac{n}{2\pi} < -\frac{1}{2} \).

Step 3: Simplify the Inequality

To isolate \( n \), multiply both sides of the inequality by \( 2\pi \):\[ n < -\pi \].Since \( n \) is an integer, and \( \pi \approx 3.14159 \), this inequality implies that the integer values for \( n \) must be less than \(-3.14159\). The largest integer less than \( -3.14159 \) is \(-4\). Thus, the maximum integer \( n \) is \(-4\).

Step 4: Find the Minimum Integer Value

For the minimum integer, note the inequality from Step 3 still holds:\[ n < -\pi \].If we are looking for the minimum value of \( n \) that still satisfies the inequality where \( n < -\pi \), then testing nearby consecutive integers gives us that the minimum integer \( n \) that satisfies this inequality is \(-6\), because it is less than \(-\pi\).

Step 5: Determine Min and Max Values

Based on the analysis, the minimum possible integer value for \( n \) is \(-6\) and the maximum is \(-4\). Therefore, solution matches option (A) \(-6\) and \(-4\).

Key Concepts

Inequalities

Inequalities in mathematics are expressions that look into different sizes and positions rather than exact values. They express relationships between quantities using symbols like ">" and "<". In the given problem, we are dealing with an inequality involving the inverse cosine function, represented by \( \cos^{-1}\left(\frac{n}{2\pi}\right) > \frac{2\pi}{3} \). This inequality tells us about a relationship between \( n \) and the angle \( \frac{2\pi}{3} \). To solve it, we need to find where our expression exceeds \( \frac{2\pi}{3} \).

In practical terms:

• Understand which range is being discussed, i.e., which values satisfy the inequality.
• Translate these into understandable comparisons, which demands knowledge of how inverse trigonometric functions behave.

The core idea is about knowing how the inequality affects \( n \), leading to finding integer solutions that fit this set range.

Integer Solutions

When dealing with inequalities and equations in mathematics, often we seek integer solutions — whole number values that satisfy the condition. Here, our inequality \( \frac{n}{2\pi} < -\frac{1}{2} \) must hold true for an integer \( n \). This translates to finding an \( n \) that is less than \( -\pi \) due to the calculations mentioned in the exercise.

To effectively find suitable integers:

• Convert mathematical expressions into simpler forms where possible, like multiplying or dividing both sides by constants.
• Remember approximation values (such as \( \pi \approx 3.14159 \)) when comparing against inequalities.
• Consider the range of numbers that satisfy the inequality — in this case, numbers smaller than \(-3.14159\).

By following this process, you determine that \( n = -6 \) and \( n = -4 \) are valid integer solutions for this context.

Cosine Function

The cosine function, denoted as \( \cos(\theta) \), is a fundamental part of trigonometry and represents the adjacent side over the hypotenuse in a right triangle. The inverse, \( \cos^{-1}(x) \), finds the angle \( \theta \) given a cosine value \( x \). In trigonometry, the range of \( \cos^{-1} \) is \([0, \pi]\). This means that whatever the angle calculated is, it will lie between these measurements.

Key points to remember about the cosine function and its inverse:

• Useful in forming relationships between angles and lengths in triangles.
• Inverse functions turn values back into angles, which can be used to solve inequalities as seen in the exercise.
• With function outputs like \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \), you can compare and solve inequalities for integer solutions.

Understanding these core principles equips you to handle problems involving the cosine function and its inverse which are common in trigonometry homework.