To find the values of [tex]\( x \)[/tex] for which [tex]\(\cos x = 0.7252\)[/tex] in the interval [tex]\(0^{\circ} \leq x \leq 360^{\circ}\)[/tex], follow these steps:
1. Determine the principal angle:
- Use the inverse cosine function to find the angle whose cosine is [tex]\(0.7252\)[/tex].
- This gives us the principal angle, which lies between [tex]\(0^{\circ}\)[/tex] and [tex]\(180^{\circ}\)[/tex].
2. Calculate the angle:
- The cosine of an angle in degrees that gives [tex]\(0.7252\)[/tex] is approximately [tex]\(44^{\circ}\)[/tex].
3. Find the second angle:
- Recall that the cosine function is positive in the first and fourth quadrants. Therefore, there is another angle in the fourth quadrant that also satisfies the given cosine value.
- The second angle can be found using the symmetry property of the cosine function:
[tex]\[
x = 360^\circ - \text{principal angle}
\][/tex]
4. Calculate the second angle in the interval:
- Using the principal angle of [tex]\(44^{\circ}\)[/tex], the second angle is:
[tex]\[
360^\circ - 44^\circ = 316^\circ
\][/tex]
5. Rounding:
- Both angles are already given to the nearest degree.
Thus, the angles [tex]\( x \)[/tex] in the interval [tex]\(0^{\circ} \leq x \leq 360^{\circ}\)[/tex] for which [tex]\(\cos x = 0.7252\)[/tex] are:
[tex]\[
\boxed{44^\circ \text{ and } 316^\circ}
\][/tex]
