Angles

(2^\circ=7200''), (20'=1200''), and the total is (8430''). Check the options carefully in such calculations.

One degree equals \(3600''\), and one minute equals \(60''\). Here, \(8430''=2\times3600''+1230''\), and \(1230''=20\times60''+30''\). Therefore, the angle is \(2^\circ\,20'\,30''\). Option B incorrectly interchanges the minutes and seconds. Exam tip: first divide total seconds by \(3600\) for degrees, then divide the remainder by \(60\) for minutes.

Convert the decimal part \(0.25^\circ\) into minutes by multiplying by \(60\): \(0.25\times 60=15'\). Therefore, \(3.25^\circ=3^\circ 15'\). The option \(3^\circ 20'\) would correspond to a decimal part of \(\frac{1}{3}^\circ\), not \(0.25^\circ\). Exam tip: multiply the decimal part of degrees by \(60\) to convert it into minutes.

(18'=\frac{18}{60}^\circ=0.30^\circ), so (6^\circ18'=6.30^\circ). Do not treat minutes directly as decimals.

Subtract one complete revolution, \(360^\circ\), from \(370^\circ\): \(370^\circ-360^\circ=10^\circ\). Thus, \(370^\circ\) is coterminal with \(10^\circ\), so its principal angle in the given interval is \(10^\circ\). An option such as \(20^\circ\) would require a remainder of \(20^\circ\), which is not obtained here. Exam tip: For angles greater than \(360^\circ\), divide by \(360^\circ\) and use the remainder to find the principal angle.

To find the principal angle, subtract complete multiples of \(360^\circ\) from the given angle. Since \(725^\circ=2\times360^\circ+5^\circ\), its principal angle is \(5^\circ\). The angle \(15^\circ\) would result from subtracting \(710^\circ\), which is not a multiple of \(360^\circ\). Exam tip: divide the angle by \(360^\circ\) and take the remainder.

Coterminal angles differ by an integer multiple of \(360^\circ\). Here, \(-210^\circ+360^\circ=150^\circ\), and \(150^\circ\) lies between \(0^\circ\) and \(360^\circ\). Therefore, the correct answer is \(150^\circ\). \(210^\circ\) is a different angle and is not obtained by adding \(360^\circ\) to \(-210^\circ\). Exam tip: To convert a negative angle to the standard range, first add \(360^\circ\).

To find the principal angle, add a multiple of \(360^\circ\) until the result lies between \(0^\circ\) and \(360^\circ\). Here, \(-675^\circ+720^\circ=45^\circ\), where \(720^\circ=2\times360^\circ\). Therefore, the principal angle is \(45^\circ\). Although \(60^\circ\) is close, it is not coterminal with \(-675^\circ\). Exam tip: for a negative angle, keep adding \(360^\circ\).

In the second quadrant the reference angle is (180^\circ-\theta), so it is (70^\circ). A reference angle is always acute.

In the third quadrant the reference angle is (\theta-180^\circ), so it is (70^\circ). Identify the quadrant first and then apply the rule.

In the fourth quadrant the reference angle is (360^\circ-\theta), so it is (50^\circ). Take the smaller angle to the (x)-axis.

In the first quadrant the reference angle is the angle itself, so it is (28^\circ). No extra calculation is needed in the first quadrant.

(380^\circ-20^\circ=360^\circ), so they are coterminal angles. If the difference is a multiple of (360^\circ), they are coterminal.

(65^\circ-360^\circ=-295^\circ). Subtract (360^\circ) to get a negative coterminal angle.

Coterminal angles differ by an integral multiple of 360^\circ. Adding 360^\circ to -40^\circ gives \((-40^\circ+360^\circ)=320^\circ\), which is positive. In contrast, 340^\circ differs from -40^\circ by 380^\circ, not a multiple of 360^\circ. Exam tip: To find the least positive coterminal angle of a negative angle, add 360^\circ.

One complete revolution equals \(360^\circ\). Therefore, the angle for 3 complete revolutions is \(3\times360^\circ=1080^\circ\). \(720^\circ\) represents only 2 complete revolutions. Exam tip: multiply the number of complete revolutions by \(360^\circ\) to convert them into degrees.

One complete revolution equals \(360^\circ\). Therefore, the angle for \(2.5\) revolutions is \(2.5\times360^\circ=900^\circ\). \(720^\circ\) represents only 2 complete revolutions, so it is not correct. Exam tip: To convert revolutions into degrees, multiply the number of revolutions by \(360^\circ\).

(810^\circ=720^\circ+90^\circ), so it is (2) complete revolutions and (90^\circ) remaining. Separate multiples of (360^\circ).

An acute angle lies between (0^\circ) and (90^\circ), so (42^\circ) is correct. Identify angle type using its range.

An obtuse angle lies between (90^\circ) and (180^\circ), so (132^\circ) is correct. (90^\circ) and (180^\circ) themselves are not obtuse angles.