Question

which expression finds the measure of an angle that is coterminal with a 126° angle? 126° + (275n)°, for any integer n 126° + (375n)°, for any integer n 126° + (450n)°, for any integer n 126° + (720n)°, for any integer n

Explanation:

Step1: Recall coterminal angle formula

Coterminal angles are found by adding or subtracting multiples of \(360^\circ\). The general formula for coterminal angles of an angle \(\theta\) is \(\theta + 360n^\circ\), where \(n\) is any integer. But we can also have multiples of \(720^\circ\) (which is \(2\times360^\circ\)) or other multiples, but let's check the options. Wait, maybe I made a mistake. Wait, let's check each option:

Wait, the angle is \(126^\circ\). Coterminal angles are \(126^\circ+ 360n^\circ\) or \(126^\circ + 720n^\circ\) (since \(720 = 2\times360\)), etc. Wait, let's check the options:

Option 1: \(126^\circ+(275n)^\circ\) – 275 is not a multiple of 360.

Option 2: \(126^\circ+(375n)^\circ\) – 375 not multiple of 360.

Option 3: \(126^\circ+(450n)^\circ\) – 450 not multiple of 360. Wait, 450 - 360 = 90, no. Wait, wait, maybe I messed up. Wait, 720 is \(2\times360\), so \(126^\circ + 720n^\circ\) is also coterminal because 720 is a multiple of 360. Let's check the options:

Wait the fourth option is \(126^\circ+(720n)^\circ\), for any integer \(n\). Let's verify:

Coterminal angles differ by \(360^\circ k\), where \(k\) is integer. So \(126^\circ + 360n^\circ\) or \(126^\circ + 720n^\circ\) (since 720 is \(2\times360\)) are both coterminal. Wait, but let's check the options again. Wait the options are:

1. \(126^\circ + (275n)^\circ\)

1. \(126^\circ + (375n)^\circ\)

1. \(126^\circ + (450n)^\circ\)

1. \(126^\circ + (720n)^\circ\)

Wait, 720 is a multiple of 360 (720 = 2*360), so adding 720n degrees (n integer) will give coterminal angles. Let's check the other options:

450n: 450 = 360 + 90, so 126 + 450n = 126 + 360n + 90n = 216 + 360n + 90n? No, wait 450n = 360n + 90n, so 126 + 450n = 126 + 360n + 90n, which is not a multiple of 360 added to 126. Wait, maybe I made a mistake. Wait, the fourth option is \(126^\circ + 720n^\circ\). Let's see: 720n is 2360n, so it's a multiple of 360 (since 720n = 360(2n)), so \(126^\circ + 720n^\circ = 126^\circ + 360*(2n)^\circ\), which is coterminal because 2n is integer (since n is integer). So that's valid.

Wait, but let's check the other options. Let's take n=1:

For option 4: 126 + 720 = 846. 846 - 126 = 720, which is 2*360, so coterminal.

For option 3: 126 + 450 = 576. 576 - 126 = 450, which is not a multiple of 360 (450/360 = 1.25), so not coterminal.

Option 2: 126 + 375 = 501. 501 - 126 = 375, not multiple of 360.

Option 1: 126 + 275 = 401. 401 - 126 = 275, not multiple of 360.

So the only option where the added term is a multiple of 360 (720 is 2*360) is the fourth option: \(126^\circ + (720n)^\circ\), for any integer \(n\).

Wait, but let's confirm the formula. Coterminal angles are angles that have the same terminal side, so they differ by \(360^\circ k\), \(k\in\mathbb{Z}\). So \(126^\circ + 360n^\circ\) or \(126^\circ + 720n^\circ\) (since 720 is \(2\times360\)) are both coterminal. So the fourth option is correct because 720 is a multiple of 360, so adding 720n (n integer) gives coterminal angles.

Step2: Verify each option

• Option 1: \(275n\) – 275 is not a multiple of 360, so \(126 + 275n\) is not coterminal.
• Option 2: \(375n\) – 375 not multiple of 360, so no.
• Option 3: \(450n\) – 450 not multiple of 360, so no.
• Option 4: \(720n\) – 720 is \(2\times360\), so \(126 + 720n\) is coterminal (since 720n is a multiple of 360, as \(720n = 360\times2n\), and \(2n\) is integer when \(n\) is integer).

Answer:

\(126^\circ + (720n)^\circ\), for any integer \(n\) (the fourth option, assuming the options are labeled as 1 to 4, with the fourth being \(126^\circ + (720n)^\circ\), for any integer \(n\))