3. Find all solutions to the trigonometric equation: 3csc² x -4 = 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°.
Related questions
Question
![### Problem Statement
**Find all solutions to the trigonometric equation:**
\[ 3 \csc^2 x - 4 = 0 \]
### Explanation
To solve this trigonometric equation, follow these steps:
1. **Rewrite the Equation:** Bring the equation into a simpler form.
\[ 3 \csc^2 x - 4 = 0 \]
Add 4 to both sides:
\[ 3 \csc^2 x = 4 \]
2. **Isolate \(\csc^2 x\):** Divide both sides by 3:
\[ \csc^2 x = \frac{4}{3} \]
3. **Find \(\csc x\):** Take the square root of both sides:
\[ \csc x = \pm \sqrt{\frac{4}{3}} \]
\[ \csc x = \pm \frac{2}{\sqrt{3}} \]
Rationalize the denominator:
\[ \csc x = \pm \frac{2\sqrt{3}}{3} \]
4. **Convert to \(\sin x\):** Recall that \( \csc x = \frac{1}{\sin x} \). So,
\[ \sin x = \pm \frac{\sqrt{3}}{2} \]
5. **Solve for \(x\):** Find \(x\) such that \( \sin x = \frac{\sqrt{3}}{2} \) and \( \sin x = -\frac{\sqrt{3}}{2} \).
The general solutions for \( x \) where \( \sin x = \frac{\sqrt{3}}{2} \) are:
\[ x = \frac{\pi}{3} + 2k\pi \text{ or } x = \frac{2\pi}{3} + 2k\pi \]
The general solutions for \( x \) where \( \sin x = -\frac{\sqrt{3}}{2} \) are:
\[ x = \frac{4\pi}{3} + 2k\pi \text{ or } x = \frac{5\pi}{3} + 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd88595c7-c499-4476-ae94-4573188e2c46%2Fba9b8a50-74f0-4734-929b-476dba0beac3%2Fhuwqkfo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Find all solutions to the trigonometric equation:**
\[ 3 \csc^2 x - 4 = 0 \]
### Explanation
To solve this trigonometric equation, follow these steps:
1. **Rewrite the Equation:** Bring the equation into a simpler form.
\[ 3 \csc^2 x - 4 = 0 \]
Add 4 to both sides:
\[ 3 \csc^2 x = 4 \]
2. **Isolate \(\csc^2 x\):** Divide both sides by 3:
\[ \csc^2 x = \frac{4}{3} \]
3. **Find \(\csc x\):** Take the square root of both sides:
\[ \csc x = \pm \sqrt{\frac{4}{3}} \]
\[ \csc x = \pm \frac{2}{\sqrt{3}} \]
Rationalize the denominator:
\[ \csc x = \pm \frac{2\sqrt{3}}{3} \]
4. **Convert to \(\sin x\):** Recall that \( \csc x = \frac{1}{\sin x} \). So,
\[ \sin x = \pm \frac{\sqrt{3}}{2} \]
5. **Solve for \(x\):** Find \(x\) such that \( \sin x = \frac{\sqrt{3}}{2} \) and \( \sin x = -\frac{\sqrt{3}}{2} \).
The general solutions for \( x \) where \( \sin x = \frac{\sqrt{3}}{2} \) are:
\[ x = \frac{\pi}{3} + 2k\pi \text{ or } x = \frac{2\pi}{3} + 2k\pi \]
The general solutions for \( x \) where \( \sin x = -\frac{\sqrt{3}}{2} \) are:
\[ x = \frac{4\pi}{3} + 2k\pi \text{ or } x = \frac{5\pi}{3} + 2
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