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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 7.6.1c
Capitolo 7, Problema 7.6.1c

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
1. c. tan^(-1)(1/√3)

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Identify the value inside the inverse tangent function: here it is \(\frac{1}{\sqrt{3}}\).
Recall that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\) in a right triangle, so we want to find an angle \(\theta\) such that \(\tan \theta = \frac{1}{\sqrt{3}}\).
Recognize the common special angle where \(\tan \theta = \frac{1}{\sqrt{3}}\) is \(\theta = 30^\circ\) or \(\theta = \frac{\pi}{6}\) radians, based on the reference triangle with sides 1 (opposite), \(\sqrt{3}\) (adjacent), and 2 (hypotenuse).
Since the problem asks to use reference triangles in an appropriate quadrant, consider the principal value range of \(\tan^{-1}\), which is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so the angle is in the first quadrant where tangent is positive.
Conclude that the angle corresponding to \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\) is the reference angle \(\frac{\pi}{6}\) radians (or \(30^\circ\)) in the first quadrant.

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Inverse Tangent Function (arctan)

The inverse tangent function, arctan or tan⁻¹, returns the angle whose tangent is a given value. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known. The output angle is typically in the range (-π/2, π/2) or (-90°, 90°).
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Inverse Tangent

Reference Triangles

Reference triangles are right triangles drawn in a coordinate plane to help find angles and trigonometric values in different quadrants. By using the acute angle in the triangle and the signs of trigonometric functions in each quadrant, one can determine the actual angle measure.
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Introduction to Trigonometric Functions

Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each affecting the sign of sine, cosine, and tangent functions. Knowing the quadrant helps determine the correct angle corresponding to a trigonometric value, especially when using inverse functions that have restricted ranges.
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Introduction to Trigonometric Functions
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Domanda del libro di testo

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. √(x^4 + x^3)

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9. True, or false? As x→∞,

c. x = O(x+5)

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c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

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c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).

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23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.

c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?

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