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tan (x) = sin(x)

tan (x) = sin(x)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The equation shown is: \[ \tan(x) = \sin(x) \] This equation represents a trigonometric relationship where the tangent of \( x \) is equal to the sine of \( x \). To solve this equation or to analyze it further, one might consider using trigonometric identities or graphing these functions to find their points of intersection. ### Key Concepts: 1. **Tangent Function (\( \tan(x) \))**: - Defined as \(\frac{\sin(x)}{\cos(x)}\). - Periodic with a period of \(\pi\). 2. **Sine Function (\( \sin(x) \))**: - Periodic with a period of \(2\pi\). - Oscillates between \(-1\) and \(1\). ### Solving the Equation: To find solutions, consider where both functions might intersect within their respective periodic intervals. Solutions can be found using analytical methods or graphing: - **Graphical Approach**: Plot \(\tan(x)\) and \(\sin(x)\) on the same graph to find intersection points. - **Analytical Approach**: Set \(\frac{\sin(x)}{\cos(x)} = \sin(x)\) and solve for \(x\), considering the restrictions where \(\cos(x) \neq 0\). These methods highlight how trigonometric equations can be analyzed using both graphical intuition and algebraic manipulation.
Transcribed Image Text:The equation shown is: \[ \tan(x) = \sin(x) \] This equation represents a trigonometric relationship where the tangent of \( x \) is equal to the sine of \( x \). To solve this equation or to analyze it further, one might consider using trigonometric identities or graphing these functions to find their points of intersection. ### Key Concepts: 1. **Tangent Function (\( \tan(x) \))**: - Defined as \(\frac{\sin(x)}{\cos(x)}\). - Periodic with a period of \(\pi\). 2. **Sine Function (\( \sin(x) \))**: - Periodic with a period of \(2\pi\). - Oscillates between \(-1\) and \(1\). ### Solving the Equation: To find solutions, consider where both functions might intersect within their respective periodic intervals. Solutions can be found using analytical methods or graphing: - **Graphical Approach**: Plot \(\tan(x)\) and \(\sin(x)\) on the same graph to find intersection points. - **Analytical Approach**: Set \(\frac{\sin(x)}{\cos(x)} = \sin(x)\) and solve for \(x\), considering the restrictions where \(\cos(x) \neq 0\). These methods highlight how trigonometric equations can be analyzed using both graphical intuition and algebraic manipulation.
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