Question

select all that are measures of angles that are coterminal with a $-50^{circ}$ angle. $-770^{circ}$ $-530^{circ}$ $-410^{circ}$ $50^{circ}$ $310^{circ}$ $360^{circ}$ $410^{circ}$ $670^{circ}$ done

Explanation:

To determine if an angle is coterminal with \(-50^\circ\), we use the formula for coterminal angles: \( \theta + 360^\circ n \), where \( n \) is an integer. We check each angle by seeing if \( \frac{\text{angle} - (-50^\circ)}{360^\circ} \) is an integer (i.e., the difference is a multiple of \( 360^\circ \)).

Step 1: Check \(-770^\circ\)

Calculate the difference: \(-770^\circ - (-50^\circ) = -720^\circ\).
Divide by \( 360^\circ \): \( \frac{-720^\circ}{360^\circ} = -2 \) (integer). So, \(-770^\circ\) is coterminal.

Step 2: Check \(-530^\circ\)

Difference: \(-530^\circ - (-50^\circ) = -480^\circ\).
Divide by \( 360^\circ \): \( \frac{-480^\circ}{360^\circ} = -\frac{4}{3} \) (not integer). Not coterminal.

Step 3: Check \(-410^\circ\)

Difference: \(-410^\circ - (-50^\circ) = -360^\circ\).
Divide by \( 360^\circ \): \( \frac{-360^\circ}{360^\circ} = -1 \) (integer). Coterminal.

Step 4: Check \(50^\circ\)

Difference: \(50^\circ - (-50^\circ) = 100^\circ\).
Divide by \( 360^\circ \): \( \frac{100^\circ}{360^\circ} = \frac{5}{18} \) (not integer). Not coterminal.

Step 5: Check \(310^\circ\)

Difference: \(310^\circ - (-50^\circ) = 360^\circ\).
Divide by \( 360^\circ \): \( \frac{360^\circ}{360^\circ} = 1 \) (integer). Coterminal.

Step 6: Check \(360^\circ\)

Difference: \(360^\circ - (-50^\circ) = 410^\circ\).
Divide by \( 360^\circ \): \( \frac{410^\circ}{360^\circ} \approx 1.138 \) (not integer). Not coterminal.

Step 7: Check \(410^\circ\)

Difference: \(410^\circ - (-50^\circ) = 460^\circ\).
Divide by \( 360^\circ \): \( \frac{460^\circ}{360^\circ} \approx 1.277 \) (not integer). Wait, correction: \(410^\circ - (-50^\circ) = 460^\circ\)? No, wait: \(410^\circ - (-50^\circ) = 460^\circ\)? Wait, no—wait, \(410^\circ - (-50^\circ) = 460^\circ\)? Wait, no, let's recalculate: \(410 - (-50) = 460\)? Wait, no, \(410 + 50 = 460\). Then \(460 / 360 = 13/9\) (not integer). Wait, but wait, maybe I made a mistake. Wait, \(410^\circ - (-50^\circ) = 460^\circ\)? No, wait, \(410 - (-50) = 460\)? Wait, no, \(410 + 50 = 460\). Then \(460 / 360 = 13/9\) (not integer). Wait, but let's check another way: \(410 - 360 = 50\), but we need coterminal with \(-50\). Wait, maybe I messed up. Wait, let's check \(670^\circ\):

Step 8: Check \(670^\circ\)

Difference: \(670^\circ - (-50^\circ) = 720^\circ\).
Divide by \( 360^\circ \): \( \frac{720^\circ}{360^\circ} = 2 \) (integer). So \(670^\circ\) is coterminal.

Wait, let's recheck \(410^\circ\): Wait, \(410 - (-50) = 460\), which is not a multiple of 360. But wait, maybe I made a mistake with \(410^\circ\). Wait, no—wait, let's check \(310^\circ\): \(310 + 50 = 360\), which is a multiple of 360 (360*1). Correct. \(670^\circ\): \(670 + 50 = 720\), which is 360*2. Correct. \(-770^\circ\): \(-770 + 50 = -720\), which is 360*(-2). Correct. \(-410^\circ\): \(-410 + 50 = -360\), which is 360*(-1). Correct. Let's check \(410^\circ\) again: \(410 - 360 = 50\), but we need coterminal with \(-50\). Wait, \(50\) and \(-50\) are not coterminal (they differ by 100, not a multiple of 360). Wait, maybe I made a mistake with \(410^\circ\). Let's check \(670^\circ\): \(670 - 360 = 310\), \(310 - 360 = -50\). Ah! There we go. So \(670^\circ - 2*360^\circ = 670 - 720 = -50^\circ\). So \(670^\circ\) is coterminal. \(310^\circ\): \(310 - 360 = -50^\circ\). So \(310^\circ = -50^\circ + 360^\circ\). Correct. \(-410^\circ\): \(-410 + 360 = -50^\circ\). So \(-410^\circ = -50^\circ - 360^\circ\). Correct. \(-770^\circ\): \(-770 + 2*360 = -770 + 720 = -50^\circ\). Correct. Now, let's check \(410^\circ\): \(410 - 360 = 50^\circ\), which is not \(-50^\circ\). So \(410^\circ\) is not coterminal. Wait, but let's check \(410^\circ\) again: \(410 - 360 = 50\), so \(410^\circ\) is coterminal with \(50^\circ\), not \(-50^\circ\). So \(410^\circ\) is not coterminal. Now, let's check \(-530^\circ\): \(-530 + 360 = -170\), \(-170 + 360 = 190\), not \(-50\). So \(-530^\circ\) is not. \(50^\circ\): \(50 - (-50) = 100\), not multiple of 360. \(360^\circ\): \(360 - (-50) = 410\), not multiple of 360. So the coterminal angles are \(-770^\circ\), \(-410^\circ\), \(310^\circ\), \(670^\circ\). Wait, wait, let's check \(410^\circ\) again: Wait, \(410 - 360 = 50\), but \(50\) and \(-50\) are not coterminal. Wait, but let's check \(670^\circ\): \(670 - 2*360 = 670 - 720 = -50\). Correct. \(310 - 360 = -50\). Correct. \(-410 + 360 = -50\). Correct. \(-770 + 2*360 = -770 + 720 = -50\). Correct. Now, let's check \(410^\circ\): \(410 - 360 = 50\), so \(410^\circ\) is coterminal with \(50^\circ\), not \(-50^\circ\). So \(410^\circ\) is not. So the correct ones are \(-770^\circ\), \(-410^\circ\), \(310^\circ\), \(670^\circ\). Wait, but let's check \(410^\circ\) again: Wait, \(410 - (-50) = 460\), which is not a multiple of 360. \(670 - (-50) = 720\), which is 2*360. Correct. \(310 - (-50) = 360\), which is 1*360. Correct. \(-410 - (-50) = -360\), which is -1*360. Correct. \(-770 - (-50) = -720\), which is -2*360. Correct. So those four are coterminal.

Answer:

The angles coterminal with \(-50^\circ\) are:

• \(-770^\circ\)
• \(-410^\circ\)
• \(310^\circ\)
• \(670^\circ\)

(Note: Upon rechecking, \(410^\circ\) was initially miscalculated; the correct coterminal angles are \(-770^\circ\), \(-410^\circ\), \(310^\circ\), and \(670^\circ\).)