To solve the equation [tex]\(\sin(2 \theta) + \cos(\theta) = 0\)[/tex], let's proceed step by step:
### Step 1: Use a Trigonometric Identity
First, we recognize that we can use the double-angle identity for sine:
[tex]\[ \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) \][/tex]
### Step 2: Substitute the Identity into the Equation
Substitute [tex]\(\sin(2 \theta)\)[/tex] into the original equation:
[tex]\[ 2 \sin(\theta) \cos(\theta) + \cos(\theta) = 0 \][/tex]
### Step 3: Factor Out [tex]\(\cos(\theta)\)[/tex]
Factor [tex]\(\cos(\theta)\)[/tex] out of the equation:
[tex]\[ \cos(\theta) (2 \sin(\theta) + 1) = 0 \][/tex]
### Step 4: Solve the Factored Equation
Now, we have two separate equations to solve:
1. [tex]\(\cos(\theta) = 0\)[/tex]
2. [tex]\(2 \sin(\theta) + 1 = 0\)[/tex]
### Step 5: Solve [tex]\(\cos(\theta) = 0\)[/tex]
Solve the first equation:
[tex]\[ \cos(\theta) = 0 \][/tex]
The values of [tex]\(\theta\)[/tex] where this is true are:
[tex]\[ \theta = \frac{\pi}{2} + k \pi, \quad k \in \mathbb{Z} \][/tex]
For the interval [tex]\(-\pi \leq \theta < \pi\)[/tex], the solutions are:
[tex]\[ \theta = \frac{\pi}{2}, \quad \theta = -\frac{\pi}{2} \][/tex]
### Step 6: Solve [tex]\(2 \sin(\theta) + 1 = 0\)[/tex]
Solve the second equation:
[tex]\[ 2 \sin(\theta) + 1 = 0 \][/tex]
[tex]\[ 2 \sin(\theta) = -1 \][/tex]
[tex]\[ \sin(\theta) = -\frac{1}{2} \][/tex]
The solutions for [tex]\(\theta\)[/tex] where [tex]\(\sin(\theta) = -\frac{1}{2}\)[/tex] in the interval [tex]\(-\pi \leq \theta < \pi\)[/tex] are:
[tex]\[ \theta = -\frac{\pi}{6}, \quad \theta = -\frac{5\pi}{6} \][/tex]
### Step 7: Combine All Solutions
Combining all the solutions we found, we get:
[tex]\[ \theta = -\frac{5\pi}{6}, \quad -\frac{\pi}{2}, \quad -\frac{\pi}{6}, \quad \frac{\pi}{2} \][/tex]
Thus, the solutions to the equation [tex]\(\sin(2 \theta) + \cos(\theta) = 0\)[/tex] are:
[tex]\[ \theta = -\frac{5\pi}{6}, -\frac{\pi}{2}, -\frac{\pi}{6}, \frac{\pi}{2} \][/tex]
