Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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**How to Solve the Equation 2 cos(x) - 1 = 0 Using the Graph** **Graph Explanation:** The graph depicts the cosine function \( y = \cos(x) \), shown as a smooth, wavy line fluctuating between -1 and 1. Two labeled intersections indicate that \( y = \frac{1}{2} \) serves as a horizontal line intersecting the cosine curve. Key points marked on the graph include: - \( x = \frac{\pi}{3} \) - \( x = \frac{5\pi}{3} \) - \( x = -\frac{\pi}{3} \) - \( x = -\frac{5\pi}{3} \) **Explanation and Solution Steps:** 1. The equation \( 2 \cos(x) - 1 = 0 \) is equivalent to \( \cos(x) = \frac{1}{2} \). 2. Intersection points of \( y = \cos(x) \) and \( y = \frac{1}{2} \) are the solutions we seek. 3. Thus, the \( x \)-values corresponding to these intersections are the solutions to the equation. **Formulation:** - The equation \( \cos(x) = \frac{1}{2} \) gives the solution values for \( x \). - The solutions within the interval \((-2\pi, 2\pi)\) are \( x = -\frac{5\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{5\pi}{3} \). These solutions specify that the cosine curve intersects the horizontal line \( y = \frac{1}{2} \) at these points within the defined interval.
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Transcribed Image Text:**How to Solve the Equation 2 cos(x) - 1 = 0 Using the Graph** **Graph Explanation:** The graph depicts the cosine function \( y = \cos(x) \), shown as a smooth, wavy line fluctuating between -1 and 1. Two labeled intersections indicate that \( y = \frac{1}{2} \) serves as a horizontal line intersecting the cosine curve. Key points marked on the graph include: - \( x = \frac{\pi}{3} \) - \( x = \frac{5\pi}{3} \) - \( x = -\frac{\pi}{3} \) - \( x = -\frac{5\pi}{3} \) **Explanation and Solution Steps:** 1. The equation \( 2 \cos(x) - 1 = 0 \) is equivalent to \( \cos(x) = \frac{1}{2} \). 2. Intersection points of \( y = \cos(x) \) and \( y = \frac{1}{2} \) are the solutions we seek. 3. Thus, the \( x \)-values corresponding to these intersections are the solutions to the equation. **Formulation:** - The equation \( \cos(x) = \frac{1}{2} \) gives the solution values for \( x \). - The solutions within the interval \((-2\pi, 2\pi)\) are \( x = -\frac{5\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{5\pi}{3} \). These solutions specify that the cosine curve intersects the horizontal line \( y = \frac{1}{2} \) at these points within the defined interval.
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