To find the positive acute reference angle for [tex]\(\cos 298^\circ\)[/tex], we need to follow a series of steps that will help us convert the given angle into its equivalent angle between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex], and then determine the reference angle.
1. Identify the given angle:
The angle given is [tex]\(298^\circ\)[/tex].
2. Determine the equivalent angle in the interval [tex]\([0^\circ, 360^\circ]\)[/tex]:
Since [tex]\(298^\circ\)[/tex] is already within the interval [tex]\([0^\circ, 360^\circ]\)[/tex], we don't need to adjust it any further. Thus, the equivalent angle is [tex]\(298^\circ\)[/tex].
3. Determine the quadrant in which [tex]\(298^\circ\)[/tex] lies:
The interval for each quadrant is:
- First Quadrant: [tex]\(0^\circ\)[/tex] to [tex]\(90^\circ\)[/tex]
- Second Quadrant: [tex]\(90^\circ\)[/tex] to [tex]\(180^\circ\)[/tex]
- Third Quadrant: [tex]\(180^\circ\)[/tex] to [tex]\(270^\circ\)[/tex]
- Fourth Quadrant: [tex]\(270^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]
Given that [tex]\(298^\circ\)[/tex] lies between [tex]\(270^\circ\)[/tex] and [tex]\(360^\circ\)[/tex], it is in the Fourth Quadrant.
4. Determine the reference angle for [tex]\(298^\circ\)[/tex]:
The reference angle in the Fourth Quadrant is found by subtracting the angle from [tex]\(360^\circ\)[/tex]:
[tex]\[
360^\circ - 298^\circ = 62^\circ
\][/tex]
5. Determine the cosine of the given angle:
The cosine of an angle in the Fourth Quadrant is the same as the cosine of its reference angle but retains the positive or negative sign based on the quadrant. Since cosine is positive in the Fourth Quadrant, we have:
[tex]\[
\cos 298^\circ = \cos 62^\circ
\][/tex]
Therefore, expressing [tex]\(\cos 298^\circ\)[/tex] in terms of its positive acute reference angle, we get:
[tex]\[
\cos 298^\circ = \cos 62^\circ
\][/tex]
The reference angle equivalent for [tex]\(298^\circ\)[/tex] is [tex]\(62^\circ\)[/tex], and hence:
[tex]\[
\cos 298^\circ = \cos 62^\circ
\][/tex]
