To determine for which value of [tex]\( n \)[/tex] the terminal side of the angle [tex]\( 468n \)[/tex] degrees in standard position falls on the [tex]\( x \)[/tex]-axis, we need to find when the angle is either [tex]\( 0^{\circ} \)[/tex] or [tex]\( 180^{\circ} \)[/tex] after reducing the angle to be within 0 to 360 degrees.
Here are the steps to solve the problem:
1. Calculate [tex]\( 468n \)[/tex] for each given [tex]\( n \)[/tex]:
- For [tex]\( n = 4 \)[/tex], the angle is [tex]\( 468 \times 4 = 1872 \)[/tex] degrees.
- For [tex]\( n = 5 \)[/tex], the angle is [tex]\( 468 \times 5 = 2340 \)[/tex] degrees.
- For [tex]\( n = 6 \)[/tex], the angle is [tex]\( 468 \times 6 = 2808 \)[/tex] degrees.
- For [tex]\( n = 7 \)[/tex], the angle is [tex]\( 468 \times 7 = 3276 \)[/tex] degrees.
2. Reduce each angle by taking modulo 360, which gives the angle within the standard 0 to 360 degrees range.
- For [tex]\( n = 4 \)[/tex], [tex]\( 1872 \mod 360 = 192 \)[/tex] degrees.
- For [tex]\( n = 5 \)[/tex], [tex]\( 2340 \mod 360 = 180 \)[/tex] degrees.
- For [tex]\( n = 6 \)[/tex], [tex]\( 2808 \mod 360 = 288 \)[/tex] degrees.
- For [tex]\( n = 7 \)[/tex], [tex]\( 3276 \mod 360 = 36 \)[/tex] degrees.
3. Check which angle lies on the [tex]\( x \)[/tex]-axis.
For an angle to fall on the [tex]\( x \)[/tex]-axis, it must be either [tex]\( 0^{\circ} \)[/tex] or [tex]\( 180^{\circ} \)[/tex].
- For [tex]\( n = 4 \)[/tex], the reduced angle is [tex]\( 192 \)[/tex], which is not on the [tex]\( x \)[/tex]-axis.
- For [tex]\( n = 5 \)[/tex], the reduced angle is [tex]\( 180 \)[/tex], which lies on the [tex]\( x \)[/tex]-axis.
- For [tex]\( n = 6 \)[/tex], the reduced angle is [tex]\( 288 \)[/tex], which is not on the [tex]\( x \)[/tex]-axis.
- For [tex]\( n = 7 \)[/tex], the reduced angle is [tex]\( 36 \)[/tex], which is not on the [tex]\( x \)[/tex]-axis.
Therefore, the value of [tex]\( n \)[/tex] for which the angle [tex]\( 468n \)[/tex] degrees falls on the [tex]\( x \)[/tex]-axis is [tex]\( n = 5 \)[/tex].
