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Chapter 6: Problem 2

Let \(\theta \in(0, \pi / 4)\) and \(t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}\) \(t_{3}=(\cot \theta)^{\tan \theta}, t_{4}=(\cot \theta)^{c c \theta \theta}\), then (a) \(t_{1}>t_{2}>t_{3}>t_{4}\) (b) \(t_{4}>t_{3}>t_{1}>t_{2}\) (c) \(t_{2}>t_{3}>t_{1}>t_{4}\) (d) \(t_{3}>t_{1}>t_{2}>t_{4}\)

Short Answer

Expert verified
The correct answer based on analysis is not present; a reconsideration of comparisons is needed.

Step by step solution

01

Express the Terms

Start by expressing each term in terms of \( \tan(\theta) \) and \( \cot(\theta) \): - \( t_1 = (\tan \theta)^{\tan \theta} = x^x \) where \( x = \tan(\theta) \).- \( t_2 = (\tan \theta)^{\cot \theta} = x^{1/x} \).- \( t_3 = (\cot \theta)^{\tan \theta} = x^{-x^2} \).- \( t_4 = (\cot \theta)^{\cot \theta} = x^{-1} \).Let's compare the expressions considering that \( x = \tan(\theta) \) and \( 0 < \theta < \pi/4 \), meaning \( 1 > x > 0 \).
02

Compare the Exponents

Given the expressions:- \( t_1 = x^x \)- \( t_2 = x^{1/x} \)- \( t_3 = x^{-x^2} \)- \( t_4 = x^{-1} \)Note that in terms of power to \( x \), all four terms decrease strictly as the exponentials increase.
03

Evaluate Exponent Sizes

Evaluate the sizes of the exponents:- For \( t_1 = x^x \), exponent \( x \) is positive.- For \( t_2 = x^{1/x} \), exponent \( 1/x \) is positive and, for small \( x \), it is larger than \( x \).- For \( t_3 = x^{-x^2} \), exponent \(-x^2\) is negative and much smaller than \( x \) or \( 1/x \).- For \( t_4 = x^{-1} \), exponent \(-1\) is also negative but larger than \(-x^2\) for small \( x \).Thus, we see that \( t_2 > t_1 \) and \( t_4 > t_3 \). The negative exponents indicate \( t_2 > t_1 > t_4 > t_3 \).
04

Assess Inequalities

Based on comparing powers and exponents:- \( t_2 = (\tan \theta)^{\cot \theta} = x^{1/x} \) is the largest.- \( t_1 = (\tan \theta)^{\tan \theta} = x^x \) is the next largest.- \( t_4 = (\cot \theta)^{\cot \theta} = x^{-1} \).- \( t_3 = (\cot \theta)^{\tan \theta} = x^{-x^2} \) is the smallest. Therefore, the inequality is \( t_2 > t_1 > t_4 > t_3 \).
05

Choose the Correct Option

Match the inequality to the options:- The inequality found is \( t_2 > t_1 > t_4 > t_3 \).Thus, the correct answer is not exactly present among the options, more analysis and mistake rectification may be needed. However, it suggests solving the incorrect Ones or closest match.

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Key Concepts

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

Trigonometric Inequalities
Trigonometric inequalities involve comparing relationships between different trigonometric functions. In exercises like these, understanding how the values of trigonometric terms change based on the angle is critical. Here, the importance lies in identifying how functions like tangent and cotangent change as the angle \( \theta \) varies within a specific range鈥攊n this case, \( 0 < \theta < \frac{\pi}{4} \).
  • For cosine and tangent: As \( \theta \) increases, \( \tan(\theta) \) increases from 0 toward 1.
  • For cotangent: \( \cot(\theta) \) shows a decreasing trend from infinity down to 1.
Recognizing these trends helps to establish which functions dominate others when comparing values based on exponents in expressions.
Exponential Expressions
Exponential expressions, such as the ones in the exercise, require careful evaluation of the base and the exponent. In our problem, we dealt with expressions like \( t_1 = (\tan \theta)^{\tan \theta} \) and \( t_2 = (\tan \theta)^{\cot \theta} \). The base numbers鈥擻( \tan(\theta) \) or \( \cot(\theta) \)鈥攁nd their respective exponents determine the size of the expression.
  • Positive exponents indicate an exponential growth potential, whereas negative ones suggest a decreasing trend effectively.
  • Comparing \( x^x \) and \( x^{1/x} \) shows how different positions of similar bases can lead to notable differences in magnitude.
Understanding these ideas allows you to predict which terms are dominant and assist in sorting them under varying domains.
Power Functions
Power functions are an essential concept when evaluating expressions like those in the exercise. They are functions of the form \( x^n \), where \( n \) is a constant exponent. Identify how the base, \( x = \tan(\theta) \) , interacts with the exponent to understand comparative sizes.
  • Exponent comparison is vital: \( x^x \) versus \( x^{1/x} \) uses exponent size for rapid determinations of which term is larger.
  • When the exponent is a function itself, as in \( x^{-x^2} \), analysis tends to be a little more complex but necessitates understanding negative and fractional exponents.
This knowledge is foundational in solving inequalities through detailed inspection of the interacting power and base.
Trigonometry in Restricted Domains
Trigonometry in restricted domains concentrates on specific intervals for trigonometric functions, like in our scenario where \( \theta \in (0, \frac{\pi}{4}) \).
  • Within this domain, both \( \tan(\theta) \>1/x, whereas -x^2 is smaller in magnitude than either \).
  • Couter or equivalent angle pair implications on functions like tangent and cotangent must be carefully considered at these small angles.
Restricted domain analysis helps establish exact behavioral characteristics of functions, facilitating precise mapping in solving problems. Identifying how the function values relate to one another within these specific parameters goes a long way in correctly asserting which sequences of trigonometric identities hold across different inequalities.

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Most popular questions from this chapter

Assertion (A): A tower leans towards west making an angle \(\alpha\) with the vertical. The angular elevation of \(B\), the top most point of the tower, is \(\beta\) as observed from a point \(C\) due east of \(\mathrm{A}\) at a distance \(\mathrm{d}\) from \(\mathrm{A}\). If the angular elevation of \(B\) from a point due east of \(C\) at a distance \(2 d\) from \(C\) is \(\gamma\), then \(2 \tan \alpha=3 \cot \beta-\cot \gamma\) Reason \((\mathbf{R}):\) In any \(\triangle A B C\), if \(B D: D C\) \(=m: n, \angle B A D=\alpha, \angle C A D=\beta\) and \(\angle A D C\) \(=\theta\), then \((m+n) \cot \theta=m \cot \alpha-n \cot \beta\)

Which one of the following pairs is not correctly matched? \(\quad\) [NDA-2008] (a) \(\sin 2 \pi \quad: \quad \sin (-2 \pi)\) (b) tna \(45^{\circ} \quad: \quad \tan \left(-315^{\circ}\right)\) (c) \(\cot \left(\tan ^{-1} 0.5\right): \tan \left(\cot ^{-1} 0.5\right)\) (d) \(\tan 420^{\circ} \quad: \tan \left(-60^{\circ}\right)\)

If \(\sin B=\frac{1}{5} \sin (2 A+B)\), then \(\frac{\tan (A+B)}{\tan A}\) is equal to \(\quad\) (a) \(\frac{5}{3}\) (b) \(\frac{2}{3}\) (c) \(\frac{3}{2}\) (d) \(\frac{3}{5}\)

Let \(\alpha, \beta\) such that \(\pi<\alpha-\beta<3 \pi\). If \(\sin \alpha+\) \(\sin \beta=-\frac{21}{65}\) and \(\cos \alpha+\cos \beta=-\frac{27}{65}\), then the value of \(\cos \frac{\alpha-\beta}{2}\) is (a) \(\frac{6}{65}\) (b) \(\frac{3}{\sqrt{130}}\) (c) \(-\frac{3}{\sqrt{130}}\) (d) \(-\frac{6}{\sqrt{65}}\)

(b) In a \(\triangle A B C, A+B=\pi-C\) \(\tan (A+B)=\tan (\pi-C) \quad\) (Using formula) get \(\tan A+\tan B+\tan C=\tan A \tan B \tan C\) But this result is use, when \(\angle \mathrm{A}, \angle \mathrm{B}, \angle C\) are the angles of triangle.
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