Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 55

Capitolo 7, Problema 55

Use a calculator to approximate each value in decimal degrees.
θ = cot⁻¹ (-0.60724226)

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Recall that the inverse cotangent function, \(\cot^{-1}(x)\), gives an angle \(\theta\) such that \(\cot(\theta) = x\). Here, you need to find \(\theta\) where \(\cot(\theta) = -0.60724226\).

Since most calculators do not have a direct \(\cot^{-1}\) function, use the identity \(\cot(\theta) = \frac{1}{\tan(\theta)}\). Therefore, \(\theta = \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right)\).

Calculate the reciprocal of the given value: \(\frac{1}{-0.60724226}\), which will be used as the input for the \(\tan^{-1}\) function.

Use your calculator to find \(\theta = \tan^{-1}\left(\frac{1}{-0.60724226}\right)\), making sure your calculator is set to degree mode to get the answer in decimal degrees.

Interpret the result carefully: since the cotangent value is negative, the angle \(\theta\) will lie in either the second or fourth quadrant. Adjust the angle accordingly if your calculator returns a principal value outside the expected range for \(\cot^{-1}\).

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Inverse Cotangent Function (cot⁻¹)

The inverse cotangent function, cot⁻¹, returns the angle whose cotangent is a given value. It is the inverse of the cotangent function, which is the ratio of the adjacent side to the opposite side in a right triangle. Understanding its range and behavior is essential for correctly interpreting the angle.

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Percorso guidato

5:37

Introduction to Cotangent Graph

Using a Calculator for Inverse Trigonometric Functions

Calculators typically provide inverse trigonometric functions like arctan, arcsin, and arccos, but may not have a direct cot⁻¹ button. To find cot⁻¹(x), you can use the identity cot⁻¹(x) = tan⁻¹(1/x), considering the sign and quadrant to get the correct angle in degrees.

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Percorso guidato

4:45

How to Use a Calculator for Trig Functions

Converting Radians to Degrees

Inverse trigonometric functions often return angles in radians by default. To express the angle in decimal degrees, multiply the radian value by 180/π. This conversion is necessary to match the problem's requirement for decimal degrees.

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Percorso guidato

5:04

Converting between Degrees & Radians

Pratica correlata

Domanda del libro di testo

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 6]. Express solutions to four decimal places.

(arctan x)³ ― x + 2 = 0

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Domanda del libro di testo

Solve each equation for all exact solutions, in degrees.

tan θ - sec θ = 1

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Solve each equation for x.

4/3 arctan x/2 = π

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