Let's solve the trigonometric equation step-by-step:
1. Start with the given equation:
[tex]\[
\cos \theta + 1 = 1
\][/tex]
2. Subtract 1 from both sides of the equation:
[tex]\[
\cos \theta + 1 - 1 = 1 - 1
\][/tex]
3. Simplify the equation:
[tex]\[
\cos \theta = 0
\][/tex]
4. Determine the values of [tex]\(\theta\)[/tex] for which [tex]\(\cos \theta = 0\)[/tex]:
The cosine of an angle is 0 at:
[tex]\[
\theta = \frac{\pi}{2} + k\pi, \quad \text{where} \quad k \in \mathbb{Z}
\][/tex]
So, the solutions to the equation [tex]\(\cos \theta + 1 = 1\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}
\][/tex]
In specific terms of radians:
[tex]\[
\theta = \frac{(2k+1)\pi}{2}, \quad k \in \mathbb{Z}
\][/tex]
Thus, the general solutions in radians are:
[tex]\[
\theta = \boxed{\frac{(2k+1)\pi}{2}} \quad \text{or} \quad \boxed{\frac{3\pi}{2} + 2k\pi, k \in \mathbb{Z} \quad \text{or} \quad \frac{\pi}{2} + 2k\pi, k \in \mathbb{Z}}
\][/tex]
Given the earlier provided solutions, the specific values found were [tex]\(\theta = 1.5707963267948966\)[/tex] and [tex]\(\theta = 4.71238898038469\)[/tex]:
- [tex]\(1.5707963267948966\)[/tex] radians corresponds to [tex]\(\frac{\pi}{2}\)[/tex].
- [tex]\(4.71238898038469\)[/tex] radians corresponds to [tex]\(\frac{3\pi}{2}\)[/tex].
Therefore, the simplified and detailed solution set for [tex]\(\cos \theta +1 = 1\)[/tex] is:
[tex]\[
\theta = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}
\][/tex]
