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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.6.132
Capitolo 7, Problema 7.6.132

132. What is special about the functions
f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?
Explain.

Guida verificata passo dopo passo
1
First, write down the given functions clearly: \(f(x) = \arcsin\left(\frac{1}{\sqrt{x^{2} + 1}}\right)\) and \(g(x) = \arctan\left(\frac{1}{x}\right)\).
Recall the definitions and ranges of the inverse trigonometric functions \(\arcsin\) and \(\arctan\), and consider how their arguments relate to each other.
Try to express both functions in terms of a common angle or variable by using trigonometric identities. For example, consider setting \(\theta = \arctan(x)\) and rewrite the expressions inside \(f(x)\) and \(g(x)\) in terms of \(\theta\).
Use the Pythagorean identity \(1 + \tan^{2}(\theta) = \sec^{2}(\theta)\) to simplify the expressions inside the inverse functions and see if \(f(x)\) and \(g(x)\) can be related or shown to be equal up to a constant.
Conclude by explaining that the two functions are essentially complementary angles or differ by a constant, highlighting the special relationship between \(f(x)\) and \(g(x)\).

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Inverse Trigonometric Functions

Inverse trigonometric functions like arcsin and arctan reverse the process of sine and tangent, returning an angle given a ratio. Understanding their domains and ranges is essential, as these functions are defined to produce principal values within specific intervals.
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Derivatives of Other Inverse Trigonometric Functions

Function Equivalence and Simplification

Determining if two functions are equivalent often involves algebraic manipulation and trigonometric identities. Simplifying expressions like arcsin(1/√(x²+1)) and arctan(1/x) can reveal hidden relationships or show that they represent the same angle under certain conditions.
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Properties of Functions

Relationship Between Arcsin and Arctan

Arcsin and arctan are related through right triangle definitions and trigonometric identities. For example, arcsin(1/√(x²+1)) can be expressed as arctan(1/x) by considering a triangle with sides x, 1, and √(x²+1), linking these inverse functions geometrically.
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Finding Area Between Curves on a Given Interval
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