To solve the equation [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex] for [tex]\(0 \leq \theta < 360^\circ\)[/tex], we need to determine the specific angles [tex]\(\theta\)[/tex] within the given interval where the cosine function takes the value [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
1. Identify the value [tex]\(\frac{\sqrt{2}}{2}\)[/tex] on the unit circle:
The cosine of an angle in a right triangle is the adjacent side over the hypotenuse. The value [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is a common trigonometric ratio that corresponds to a known angle.
2. Recall the first quadrant angle:
In the first quadrant (where all trigonometric ratios are positive), the cosine of 45 degrees is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. Therefore:
[tex]\[
\theta_1 = 45^\circ
\][/tex]
3. Identify the symmetrical angle in another quadrant:
The cosine function is positive in the first and fourth quadrants. The reference angle in the fourth quadrant that has the same cosine value as 45 degrees is calculated by subtracting 45 degrees from 360 degrees:
[tex]\[
\theta_2 = 360^\circ - 45^\circ = 315^\circ
\][/tex]
Thus, the two angles where [tex]\(\cos(\theta) = \frac{\sqrt{2}}{2}\)[/tex] for [tex]\(0 \leq \theta < 360^\circ\)[/tex] are [tex]\(\theta_1 = 45^\circ\)[/tex] and [tex]\(\theta_2 = 315^\circ\)[/tex].
Thereby, the correct answer is:
A. [tex]\(\theta = 45^\circ\)[/tex] and [tex]\(\theta = 315^\circ\)[/tex]
