To solve [tex]\(\cos x = \frac{\sqrt{2}}{2}\)[/tex] and list the solutions from least to greatest, we first identify all possible angles within one full rotation (0 to [tex]\(2\pi\)[/tex]) for which the cosine of the angle equals [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
### Step-by-Step Solution:
1. Unit Circle Considerations:
- Recall that the cosine of an angle in the unit circle corresponds to the x-coordinate of a point on the circle.
- [tex]\(\cos x = \frac{\sqrt{2}}{2}\)[/tex] occurs at specific angles where the x-coordinate equals [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
2. Primary Angles:
- [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is a common cosine value for the angles:
- [tex]\( x = \frac{\pi}{4} \)[/tex]
- [tex]\( x = \frac{7\pi}{4} \)[/tex]
3. Periodicity Considerations:
- Because cosine has a period of [tex]\(2\pi\)[/tex], we can express these angles periodically:
- [tex]\( x = \frac{\pi}{4} + 2k\pi \)[/tex] where [tex]\(k\)[/tex] is any integer.
- [tex]\( x = \frac{7\pi}{4} + 2k\pi \)[/tex] where [tex]\(k\)[/tex] is any integer.
4. List Specific Solutions:
- To generate a set of solutions ordered from least to greatest, we specifically consider the solutions in the range [tex]\(0 \leq x < 4\pi\)[/tex].
5. Evaluate Specific Solutions:
- Primary angles:
- [tex]\(\frac{\pi}{4}\)[/tex]
- [tex]\(\frac{7\pi}{4}\)[/tex]
- Adding the period of [tex]\(2\pi\)[/tex]:
- [tex]\(2\pi + \frac{\pi}{4} = \frac{9\pi}{4}\)[/tex]
- [tex]\(2\pi + \frac{7\pi}{4} = \frac{15\pi}{4}\)[/tex]
6. Convert into Numerical Values:
- [tex]\(\frac{\pi}{4} = 0.7853981633974483\)[/tex]
- [tex]\(\frac{7\pi}{4} = 5.497787143782138\)[/tex]
- [tex]\(\frac{9\pi}{4} = 7.0685834705770345\)[/tex]
- [tex]\(\frac{15\pi}{4} = 11.780972450961723\)[/tex]
7. Order from Least to Greatest:
- Arrange the solutions in ascending order:
[tex]\[
\left[0.7853981633974483, 5.4977871437821380, 7.0685834705770345, 11.780972450961723\right]
\][/tex]
The ordered solutions are:
[tex]\[
x = 0.7853981633974483,\ 5.497787143782138,\ 7.0685834705770345,\ 11.780972450961723
\][/tex]
