Find all solutions of the equation in the interval [0, 27). sin 6x cos 2x - cos 6x sin 2x = -1 Write your answer in radians in terms of . If there is more than one solution, separate them with commas. x = 0
Find all solutions of the equation in the interval [0, 27). sin 6x cos 2x - cos 6x sin 2x = -1 Write your answer in radians in terms of . If there is more than one solution, separate them with commas. x = 0
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Find all solutions of the equation
![### Solving the Trigonometric Equation
**Problem Statement:**
Find all solutions of the equation in the interval \([0, 2\pi)\).
\[ \sin 6x \cos 2x - \cos 6x \sin 2x = -1 \]
**Instructions:**
Write your answer in [radians](https://www.mathsisfun.com/definitions/radian.html) and in terms of \(\pi\).
If there is more than one solution, separate them with commas.
**Solution Input Field:**
\[ x = \boxed{\ \ \ \ \ \ \ \ }\]
To solve the equation, we use trigonometric identities and properties of trigonometric functions within the given interval. First, recognize that the left-hand side of the equation matches the angle addition formula for sine:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
By comparing, we identify that:
\[ A = 6x \]
\[ B = 2x \]
Thus:
\[ \sin (6x - 2x) = \sin 4x = -1 \]
Next, we determine where the sine function equals \(-1\) within the interval \([0, 2\pi)\):
\[ 4x = \frac{3\pi}{2} + 2k\pi \]
Solving for \(x\):
\[ x = \frac{3\pi}{8} + \frac{k\pi}{2} \]
For \(k=0\):
\[ x = \frac{3\pi}{8} \]
For \(k=1\):
\[ x = \frac{3\pi}{8} + \frac{\pi}{2} = \frac{3\pi + 4\pi}{8} = \frac{7\pi}{8} \]
For \(k=2\):
\[ x = \frac{3\pi}{8} + \pi = \frac{3\pi + 8\pi}{8} = \frac{11\pi}{8} \]
For \(k=3\):
\[ x = \frac{3\pi}{8} + \frac{3\pi}{2} = \frac{3\pi + 12\pi}{8} = \frac{15\pi}{8} \]
For](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c0c699c-0ea3-4621-a196-a76a2f333ed8%2Ff5b1bdf6-31bd-49b6-876b-14946a16a1a0%2Fuhf2z8s_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving the Trigonometric Equation
**Problem Statement:**
Find all solutions of the equation in the interval \([0, 2\pi)\).
\[ \sin 6x \cos 2x - \cos 6x \sin 2x = -1 \]
**Instructions:**
Write your answer in [radians](https://www.mathsisfun.com/definitions/radian.html) and in terms of \(\pi\).
If there is more than one solution, separate them with commas.
**Solution Input Field:**
\[ x = \boxed{\ \ \ \ \ \ \ \ }\]
To solve the equation, we use trigonometric identities and properties of trigonometric functions within the given interval. First, recognize that the left-hand side of the equation matches the angle addition formula for sine:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
By comparing, we identify that:
\[ A = 6x \]
\[ B = 2x \]
Thus:
\[ \sin (6x - 2x) = \sin 4x = -1 \]
Next, we determine where the sine function equals \(-1\) within the interval \([0, 2\pi)\):
\[ 4x = \frac{3\pi}{2} + 2k\pi \]
Solving for \(x\):
\[ x = \frac{3\pi}{8} + \frac{k\pi}{2} \]
For \(k=0\):
\[ x = \frac{3\pi}{8} \]
For \(k=1\):
\[ x = \frac{3\pi}{8} + \frac{\pi}{2} = \frac{3\pi + 4\pi}{8} = \frac{7\pi}{8} \]
For \(k=2\):
\[ x = \frac{3\pi}{8} + \pi = \frac{3\pi + 8\pi}{8} = \frac{11\pi}{8} \]
For \(k=3\):
\[ x = \frac{3\pi}{8} + \frac{3\pi}{2} = \frac{3\pi + 12\pi}{8} = \frac{15\pi}{8} \]
For
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