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 47

Capitolo 7, Problema 47

Find the degree measure of θ if it exists. Do not use a calculator.
θ = sin⁻¹ 2

Guida verificata passo dopo passo

Recall the definition of the inverse sine function, $\sin^{-1} x$, which gives the angle $\theta$ such that $\sin \theta = x$ and $\theta$ lies within the range $[-90^\circ, 90^\circ]$ (or $[-\frac{\pi}{2}, \frac{\pi}{2}]$$).

Understand the domain of the sine function: since $\sin \theta$ can only take values between $-1$ and $1$ for real angles $\theta$, the input to $\sin^{-1}$ must be within this range.

Check the given value inside the inverse sine: here, it is $2$, which is outside the domain $[-1, 1]$.

Since $2$ is not in the domain of the inverse sine function, there is no real angle $\theta$ such that $\sin \theta = 2$.

Therefore, conclude that $\theta = \sin^{-1} 2$ does not exist as a real number (no degree measure satisfies this).

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Domain of the Sine Function

The sine function outputs values only between -1 and 1 for real angles. Any input outside this range, such as 2, is not possible for sine, meaning sin⁻¹(2) does not correspond to a real angle.

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Percorso guidato

3:43

Finding the Domain of an Equation

Inverse Sine Function (Arcsin)

The inverse sine function, sin⁻¹(x), returns the angle whose sine is x, but only if x is within the domain [-1, 1]. It is used to find angles from sine values, but it is undefined for values outside this range.

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Percorso guidato

4:03

Inverse Sine

Understanding Restrictions in Trigonometric Functions

Trigonometric functions have specific domains and ranges that limit their outputs and inputs. Recognizing these restrictions helps determine whether an expression like sin⁻¹(2) has a valid solution or not.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

Use a calculator to approximate each value in decimal degrees.

θ = arccos (-0.39876459)

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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 exact solutions.

arcsin 2x + arccos x = π/6

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Domanda del libro di testo

Use a calculator to approximate each value in decimal degrees.

θ = sin⁻¹ (-0.13349122)

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