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Chapter 4: Problem 25

Complete set of values of \(x\) satisfying \(\left[\tan ^{-1} x\right]+\left[\cot ^{-1} x\right]\) \(=2\) where [.] denotes the greatest integer function, is (a) \((\cot 2, \cot 3)\) (b) \((\cot 3,-\tan 1)\) (c) \((\cot 3,0)\) (d) None of these

Short Answer

Expert verified
The correct answer is (d) None of these, as the range of values for \(x\) satisfying the given condition is \(\tan 1 < x < \cot 3\), and none of the provided options include only the values in that range.

Step by step solution

01

Identifying the relationship between the inverse functions

Let's first identify the relationship between \(\tan^{-1}x\) and \(\cot^{-1}x\). We have the following relationship: \(\cot^{-1}x = \frac{\pi}{2} - \tan^{-1}x\).
02

Express cotangent inverse in terms of tangent inverse

Using the relationship we found in Step 1, we can express the original equation in terms of tangent inverse function only: \(\left[\tan^{-1}x\right] + \left[\frac{\pi}{2} - \tan^{-1}x\right] = 2\).
03

Find the range of tangent inverse for the given condition

Since the sum of these two greatest integer functions is equal to 2, and both greatest integer functions are positive, we can deduce that one of them has to be equal to 1 and the other equal to 1. Considering the range of tangent inverse function, we find the following constraints: \(1 \leq \tan^{-1}x < \frac{\pi}{2}\) and \(1 \leq \frac{\pi}{2} - \tan^{-1}x < 2\).
04

Solve the constraints and find values of x

To find the values of \(x\), let's first solve the inequalities obtained in Step 3: From the first inequality: \(1 \leq \tan^{-1}x < \frac{\pi}{2}\), we find: \(x > \tan 1\). From the second inequality: \(1 \leq \frac{\pi}{2} - \tan^{-1}x < 2\), we can rewrite it as: \(\frac{\pi}{2} - 2 < \tan^{-1}x < \frac{\pi}{2} - 1\), which simplifies to: \(x < \cot 3\). Combining both constraints, we get the possible range of values for \(x\): \(\tan 1 < x < \cot 3\). Looking at the options provided, we see that none of the given alternatives include only the values in that range. So, the correct answer is: (d) None of these

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Tangent Inverse Function
The tangent inverse function, denoted as \(\tan^{-1}x\), is a way to find an angle whose tangent is a given number. This function is the inverse of the tangent function in trigonometry. When we talk about \(\tan^{-1}x\), we're looking for an angle \(\theta\) such that \(\tan\theta = x\).

The range of the tangent inverse function is crucial. It lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), meaning it can only produce angles within this interval. This makes it helpful for translating a tangent value back into an angle.

For example, if \(\tan\theta = 1\), then \(\theta\) would be \(\frac{\pi}{4}\) because \(\tan\left(\frac{\pi}{4}\right) = 1\). Understanding the limits and behavior of the tangent inverse function helps solve equations involving it, especially when paired with other trigonometric functions.
Insight into the Cotangent Inverse Function
The cotangent inverse function, represented as \(\cot^{-1}x\), determines the angle whose cotangent is \(x\). Just like the tangent inverse function, it's the inverse of the cotangent. If \(\cot\theta = x\), then \(\theta = \cot^{-1}x\).

The relationship of \(\cot^{-1}x\) with \(\tan^{-1}x\) is vital: \(\cot^{-1}x = \frac{\pi}{2} - \tan^{-1}x\). This connection helps us transform complex problems into simpler ones.Using this, you can convert expressions involving \(\cot^{-1}x\) into those involving \(\tan^{-1}x\), which might be simpler to work with.

For instance, if \(x = 1\), knowing \(\tan^{-1}1 = \frac{\pi}{4}\) lets us deduce \(\cot^{-1}1 = \frac{\pi}{4}\). Having these angles helps isolate "x" in particular equations, especially when paired with greatest integer functions.
Explaining the Greatest Integer Function
The greatest integer function, often denoted as \([\cdot]\), assigns to any real number the greatest integer less than or equal to that number. It's like rounding down to the nearest whole number.

For example, if we have \(x = 3.8\), then \([x] = 3\), and if \(x = -1.3\), then \([x] = -2\). This function is extremely helpful in mathematical problems where values are constrained to remain within integer limits.

The greatest integer function appears frequently in trigonometric problems, especially when dealing with sums of inverses like in our exercise. It is used to simplify expressions and solve problems involving ranges or intervals of functions like \(\tan^{-1}\) and \(\cot^{-1}\).

By combining these functions, we refine our solutions because the greatest integer function restricts the outcome to specific values, making it easier to determine specific ranges or solutions in a problem.

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Most popular questions from this chapter

The sum of infinite terms of the series \(\cot ^{-1}\left(2^{2}+\frac{1}{2}\right)+\) \(\cot ^{-1}\left(2^{3}+\frac{1}{2^{2}}\right)+\cot ^{-1}\left(2^{4}+\frac{1}{2^{3}}\right)+\ldots .\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{2}\) (c) \(\cot ^{-1} 2\) (d) \(-\cot ^{-1} 2\)

If \(\mathrm{f}(\mathrm{x})=\cos ^{-1}(\cos \mathrm{x})-\sin ^{-1}(\sin \mathrm{x})\); then the area bounded by \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{x}\)-axis on -interval \([-6 \pi, 6 \pi]\) is (a) \(12 \pi^{2}\) (b) \(3 \pi^{2}\) (c) \(6 \pi^{2}\) (d) None of these

The value of \(\sin ^{-1} x+\cos ^{-1} x\) is equal to (a) \(\frac{\pi}{2}\) (b) \(\frac{3 \pi}{2}\) (c) \(\frac{5 \pi}{2}\) (d) \(\frac{7 \pi}{2}\)

\(\mathrm{f}(\mathrm{k})=k^{2}\) \(\Rightarrow\) range of \(\mathrm{f}(\mathrm{k})=[0, \infty)\) But \(k \in \mathbb{N}\) \(\Rightarrow\) Range of \(\mathrm{f}(\mathrm{k})=\\{1,4,9,16, \ldots .\\}\) Ans (d)

If \(x>1\), then the value of \(2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{2}\) (c) \(\pi\) (d) \(\frac{3 \pi}{2}\)
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