To find the values of [tex]\( x \)[/tex] in the interval [tex]\( 0^{\circ} < x < 360^{\circ} \)[/tex] for which [tex]\( \cos x = -0.4226 \)[/tex], follow these steps:
1. Identify the cosine function properties:
- The cosine function is negative in the second and third quadrants.
- We need to find the reference angle first.
2. Determine the reference angle:
- The reference angle is the angle in the first quadrant for which the cosine value is the absolute value of the given cosine.
- For [tex]\(\cos x = -0.4226\)[/tex], find the angle [tex]\(\theta\)[/tex] satisfying [tex]\(\cos \theta = 0.4226\)[/tex].
3. Find the reference angle:
- The reference angle in degrees is approximately [tex]\( 65.001^\circ \)[/tex].
4. Calculate the angles in the second and third quadrants:
- In the second quadrant, the angle is [tex]\( 180^\circ - \text{reference angle} \)[/tex].
[tex]\[
\text{Angle in second quadrant} = 180^\circ - 65.001^\circ \approx 115^\circ
\][/tex]
- In the third quadrant, the angle is [tex]\( 180^\circ + \text{reference angle} \)[/tex].
[tex]\[
\text{Angle in third quadrant} = 180^\circ + 65.001^\circ \approx 245^\circ
\][/tex]
5. Round the angles to the nearest degree:
- The angles are [tex]\( 115^\circ \)[/tex] and [tex]\( 245^\circ \)[/tex].
### Final Answer:
The values of [tex]\( x \)[/tex] for which [tex]\( \cos x = -0.4226 \)[/tex] in the interval [tex]\( 0^\circ < x < 360^\circ \)[/tex] are:
[tex]\[
\boxed{115^\circ \text{ and } 245^\circ}
\][/tex]
