Find all solutions of the equation in the interval [0,2л). 2 sin ²0-sin 0-1=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°.
icon
Related questions
Question

Show the steps needed to solve this

### Problem Statement

Find all solutions of the equation in the interval \([0, 2\pi]\).

\[2 \sin^2 \theta - \sin \theta - 1 = 0\]

### Explanation

This equation is quadratic in form, where the variable is \(\sin \theta\). Solving the equation involves finding the values of \(\theta\) for which the equation holds true within the specified interval.

### Solving the Equation

1. **Substitution:**
   Let \(x = \sin \theta\). The equation becomes:
   \[2x^2 - x - 1 = 0\]

2. **Factoring:**
   Factor the quadratic equation:
   \[(2x + 1)(x - 1) = 0\]

3. **Finding Solutions:**
   Solve for \(x\):
   \[
   \begin{align*}
   2x + 1 = 0 & \quad \Rightarrow \quad x = -\frac{1}{2} \\
   x - 1 = 0 & \quad \Rightarrow \quad x = 1
   \end{align*}
   \]

4. **Determine θ Values:**

   - **For \(x = -\frac{1}{2}\):**
     Solve \(\sin \theta = -\frac{1}{2}\) in \([0, 2\pi]\).
     \[
     \theta = \frac{7\pi}{6}, \frac{11\pi}{6}
     \]

   - **For \(x = 1\):**
     Solve \(\sin \theta = 1\) in \([0, 2\pi]\).
     \[
     \theta = \frac{\pi}{2}
     \]

### Solutions

Thus, the solutions to the equation \(2 \sin^2 \theta - \sin \theta - 1 = 0\) in the interval \([0, 2\pi]\) are:
\[ \theta = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} \]

These values of \(\theta\) satisfy the equation within the given interval.
Transcribed Image Text:### Problem Statement Find all solutions of the equation in the interval \([0, 2\pi]\). \[2 \sin^2 \theta - \sin \theta - 1 = 0\] ### Explanation This equation is quadratic in form, where the variable is \(\sin \theta\). Solving the equation involves finding the values of \(\theta\) for which the equation holds true within the specified interval. ### Solving the Equation 1. **Substitution:** Let \(x = \sin \theta\). The equation becomes: \[2x^2 - x - 1 = 0\] 2. **Factoring:** Factor the quadratic equation: \[(2x + 1)(x - 1) = 0\] 3. **Finding Solutions:** Solve for \(x\): \[ \begin{align*} 2x + 1 = 0 & \quad \Rightarrow \quad x = -\frac{1}{2} \\ x - 1 = 0 & \quad \Rightarrow \quad x = 1 \end{align*} \] 4. **Determine θ Values:** - **For \(x = -\frac{1}{2}\):** Solve \(\sin \theta = -\frac{1}{2}\) in \([0, 2\pi]\). \[ \theta = \frac{7\pi}{6}, \frac{11\pi}{6} \] - **For \(x = 1\):** Solve \(\sin \theta = 1\) in \([0, 2\pi]\). \[ \theta = \frac{\pi}{2} \] ### Solutions Thus, the solutions to the equation \(2 \sin^2 \theta - \sin \theta - 1 = 0\) in the interval \([0, 2\pi]\) are: \[ \theta = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} \] These values of \(\theta\) satisfy the equation within the given interval.
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

Blurred answer
Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Algebra and Trigonometry
Algebra and Trigonometry
Trigonometry
ISBN:
9781938168376
Author:
Jay Abramson
Publisher:
OpenStax
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning