Find all solutions for cach equation in the interval (0", 360"). 31. 4cose-2/3= 0
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 5GP
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![**Trigonometric Equations: Solving for Angles in a Given Interval**
**Problem Statement:**
Find all solutions for each equation in the interval \([0^\circ, 360^\circ)\).
**Example Equation:**
31. \( 4 \cos \theta - 2 \sqrt{3} = 0 \)
**Explanation:**
We need to solve the given trigonometric equation for \(\theta\) within the interval ranging from \(0^\circ\) to \(360^\circ\).
Steps to Solve:
1. Isolate the trigonometric function.
\[ 4 \cos \theta - 2 \sqrt{3} = 0 \]
\[ 4 \cos \theta = 2 \sqrt{3} \]
\[ \cos \theta = \frac{2 \sqrt{3}}{4} \]
\[ \cos \theta = \frac{\sqrt{3}}{2} \]
2. Determine the standard angles within one period for which \(\cos \theta = \frac{\sqrt{3}}{2}\).
The cosine function \(\cos \theta = \frac{\sqrt{3}}{2}\) corresponds to these angles in the unit circle:
\[ \theta = 30^\circ \]
\[ \theta = 330^\circ \]
Thus, the solutions to the equation \(4 \cos \theta - 2 \sqrt{3} = 0\) on the interval \([0^\circ, 360^\circ)\) are:
\[ \theta = 30^\circ \]
\[ \theta = 330^\circ \]
**Graphical Representation:**
For those referring to a unit circle or a cosine graph, note that the cosine function peaks at \(1\) (or \(\cos 0^\circ = 1\)) and attains \(\frac{\sqrt{3}}{2}\) at \(30^\circ\) and \(330^\circ\). It can be helpful to sketch the unit circle or the graph of \( y = \cos \theta \) to visualize these angles and ensure the correctness of solutions.
Feel free to apply these steps to solve similar trigonometric equations within specified intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2107d902-1438-4d77-9326-6ff7a3e6e343%2Fb9d0045f-ee6e-490d-87e6-e29ad848ba14%2Fcdn70j1m_processed.png&w=3840&q=75)
Transcribed Image Text:**Trigonometric Equations: Solving for Angles in a Given Interval**
**Problem Statement:**
Find all solutions for each equation in the interval \([0^\circ, 360^\circ)\).
**Example Equation:**
31. \( 4 \cos \theta - 2 \sqrt{3} = 0 \)
**Explanation:**
We need to solve the given trigonometric equation for \(\theta\) within the interval ranging from \(0^\circ\) to \(360^\circ\).
Steps to Solve:
1. Isolate the trigonometric function.
\[ 4 \cos \theta - 2 \sqrt{3} = 0 \]
\[ 4 \cos \theta = 2 \sqrt{3} \]
\[ \cos \theta = \frac{2 \sqrt{3}}{4} \]
\[ \cos \theta = \frac{\sqrt{3}}{2} \]
2. Determine the standard angles within one period for which \(\cos \theta = \frac{\sqrt{3}}{2}\).
The cosine function \(\cos \theta = \frac{\sqrt{3}}{2}\) corresponds to these angles in the unit circle:
\[ \theta = 30^\circ \]
\[ \theta = 330^\circ \]
Thus, the solutions to the equation \(4 \cos \theta - 2 \sqrt{3} = 0\) on the interval \([0^\circ, 360^\circ)\) are:
\[ \theta = 30^\circ \]
\[ \theta = 330^\circ \]
**Graphical Representation:**
For those referring to a unit circle or a cosine graph, note that the cosine function peaks at \(1\) (or \(\cos 0^\circ = 1\)) and attains \(\frac{\sqrt{3}}{2}\) at \(30^\circ\) and \(330^\circ\). It can be helpful to sketch the unit circle or the graph of \( y = \cos \theta \) to visualize these angles and ensure the correctness of solutions.
Feel free to apply these steps to solve similar trigonometric equations within specified intervals.
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