Question

find the measure (in degrees, not equal to the given measure) of the least positive angle that is coterminal with a. a = 189° the measure of the least positive coterminal angle is \\(\square\\)°. (simplify your answer.)

Explanation:

Step1: Recall coterminal angle formula

To find a coterminal angle with an angle \( \theta \) in degrees, we use the formula \( \theta + 360^{\circ}n \), where \( n \) is an integer. We need the least positive angle not equal to the given angle, so we try \( n = 1 \) (adding \( 360^{\circ} \)) or \( n=- 1 \) (subtracting \( 360^{\circ} \)) and see which gives a positive angle.

Step2: Calculate for \( n = 1 \)

For \( \theta=189^{\circ} \) and \( n = 1 \), the angle is \( 189^{\circ}+360^{\circ}\times1=189^{\circ} + 360^{\circ}=549^{\circ} \). But we can also check \( n=- 1 \), but \( 189^{\circ}-360^{\circ}=- 171^{\circ} \) which is negative. So the least positive coterminal angle (not equal to \( 189^{\circ} \)) is obtained by adding \( 360^{\circ} \) to \( 189^{\circ} \). Wait, no, wait. Wait, coterminal angles can be found by adding or subtracting multiples of \( 360^{\circ} \). Wait, the given angle is \( 189^{\circ} \), to find the least positive coterminal angle not equal to it, we add \( 360^{\circ} \) (since subtracting \( 360^{\circ} \) gives a negative angle). So \( 189^{\circ}+360^{\circ}=549^{\circ} \)? Wait, no, that's not right. Wait, maybe I made a mistake. Wait, coterminal angles: two angles are coterminal if their difference is a multiple of \( 360^{\circ} \). So for angle \( A = 189^{\circ} \), to find the least positive coterminal angle not equal to \( 189^{\circ} \), we can add \( 360^{\circ} \) (because subtracting \( 360^{\circ} \) gives \( 189 - 360=-171^{\circ} \), which is negative). So \( 189+360 = 549^{\circ} \)? Wait, but let's check. Wait, maybe the problem is to find the coterminal angle by adding or subtracting \( 360^{\circ} \) to get the least positive one different from the original. So original angle is \( 189^{\circ} \), adding \( 360^{\circ} \) gives \( 189 + 360=549^{\circ} \), which is positive and not equal to \( 189^{\circ} \). Subtracting \( 360^{\circ} \) gives \( - 171^{\circ} \), which is negative, so we discard that. So the least positive coterminal angle is \( 549^{\circ} \)? Wait, no, wait, maybe I messed up. Wait, coterminal angles: the formula is \( \theta + 360k \), where \( k \) is integer. We need the smallest positive angle not equal to \( \theta \). So for \( \theta = 189^{\circ} \), when \( k = 1 \), \( 189+360 = 549^{\circ} \), when \( k=- 1 \), \( 189 - 360=-171^{\circ} \) (negative). So the least positive one is \( 549^{\circ} \)? Wait, but let's confirm. Let's think again. The coterminal angles are angles that share the same terminal side. So to find a positive coterminal angle different from \( 189^{\circ} \), we add \( 360^{\circ} \) (since subtracting would give a negative angle). So \( 189 + 360=549 \). So that's the least positive coterminal angle not equal to \( 189^{\circ} \).

Answer:

\( 549 \)