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An angle measuring [tex]\( 468n^\circ \)[/tex] is in standard position. For which value of [tex]\( n \)[/tex] will the terminal side fall on the [tex]\( x \)[/tex]-axis?

A. [tex]\( n = 4 \)[/tex]
B. [tex]\( n = 5 \)[/tex]
C. [tex]\( n = 6 \)[/tex]
D. [tex]\( n = 7 \)[/tex]

Sagot :

GinnyAnswer
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].
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