Question

find a positive angle less than 360° that is coterminal with the given angle. 670° a positive angle less than 360° that is coterminal with 670° is □°.

Explanation:

Step1: Recall coterminal angle formula

To find a positive coterminal angle less than \(360^\circ\) for an angle \(\theta\), we subtract multiples of \(360^\circ\) until the result is positive and less than \(360^\circ\). The formula is \(\theta - n\times360^\circ\), where \(n\) is a positive integer such that the result is in the desired range.

Step2: Determine the multiple of \(360^\circ\) to subtract

For \(\theta = 670^\circ\), we need to find \(n\) such that \(670^\circ - n\times360^\circ\) is positive and less than \(360^\circ\). Let's try \(n = 1\): \(670 - 360 = 310^\circ\). Wait, no, wait, \(670 - 2\times360 = 670 - 720=- 50\), which is negative. Wait, no, I made a mistake. Wait, \(360\times1 = 360\), \(670 - 360 = 310\)? Wait, no, \(360\times2 = 720\), which is more than 670. Wait, no, \(670\div360\approx1.86\), so we take \(n = 1\)? Wait, no, \(670 - 360 = 310\)? Wait, no, \(360\times1 = 360\), \(670-360 = 310\)? Wait, but \(310\) is less than \(360\)? Wait, no, \(360\times1 = 360\), \(670 - 360 = 310\), which is positive and less than \(360\)? Wait, no, \(360\times2 = 720\), which is more than 670, so \(n = 1\) gives \(670 - 360 = 310\)? Wait, no, that can't be. Wait, no, \(670 - 360 = 310\), which is less than \(360\), so that's the coterminal angle. Wait, but let's check: coterminal angles differ by multiples of \(360^\circ\). So \(670^\circ - 360^\circ=310^\circ\), and \(310^\circ + 360^\circ = 670^\circ\), so they are coterminal.

Wait, I think I messed up earlier. Let's do it correctly. The formula for coterminal angles is \(\theta + k\times360^\circ\) or \(\theta - k\times360^\circ\) where \(k\) is an integer. To get a positive angle less than \(360^\circ\) for \(\theta = 670^\circ\), we subtract \(360^\circ\) once: \(670 - 360 = 310^\circ\). Wait, but \(310\) is less than \(360\), so that's the angle. Wait, but let's verify: \(310 + 360 = 670\), so yes, they are coterminal.

Answer:

\(310\)