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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.61
Capitolo 7, Problema 6.61

Solve each equation for x.
y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]

Guida verificata passo dopo passo
1
Start by isolating the tangent function: multiply both sides of the equation by 2 to get \( \tan(3x + 2) = 2y \).
Next, take the inverse tangent (arctan) of both sides to solve for the angle: \( 3x + 2 = \arctan(2y) \).
Subtract 2 from both sides to isolate the term with x: \( 3x = \arctan(2y) - 2 \).
Divide both sides by 3 to solve for x: \( x = \frac{\arctan(2y) - 2}{3} \).
Ensure that the solution for x falls within the given interval \([-\frac{2}{3} - \frac{\pi}{6}, -\frac{2}{3} + \frac{\pi}{6}]\) by checking the calculated x values.

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Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning it repeats its values every π radians. Understanding the properties of the tangent function, including its asymptotes and behavior, is crucial for solving equations involving tan.
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Introduction to Tangent Graph

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find angles when the value of a trigonometric function is known. For example, if y = tan(x), then x = arctan(y). These functions are essential for solving equations where the variable is inside a trigonometric function, allowing us to isolate the angle and find the corresponding x values.
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Introduction to Inverse Trig Functions

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [-2/3 - π/6, -2/3 + π/6] specifies the domain within which we are looking for solutions for x. Understanding how to interpret and work within specified intervals is important for determining valid solutions to trigonometric equations.
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i & j Notation
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