Angles

At (0^\circ), the terminal side stays with the initial side on the positive (x)-axis. A zero angle has no rotation.

(-90^\circ) is clockwise rotation so the terminal side lies on the negative (y)-axis. The negative sign shows direction.

There are 60 minutes in 1 degree. Thus, \(25^\circ=25\times60'=1500'\); adding the remaining \(30'\) gives \(1500'+30'=1530'\). \(1500'\) represents only 25 degrees and does not include the extra 30 minutes. Exam tip: To convert a mixed angle into minutes, multiply the degrees by 60 and then add the minutes.

The governing conversion is based on the fact that one degree contains 60 minutes. The apostrophe indicates minutes, so 45' must be converted by dividing by 60, not by treating 45 as a decimal digit. Thus 45 minutes equals 45/60 degree, which is 0.75°. Adding this fractional part to the whole-degree measure gives 12°45' = 12 + 45/60 = 12 + 0.75 = 12.75°. Therefore option B is correct. Option A incorrectly uses 0.25° for the minutes, option C reflects an inaccurate conversion, and option D makes the minute contribution too large. The same method works for any degree-minute conversion: divide the minutes by 60 and then add the result to the degree value.

Convert the decimal part 0.5° into minutes by multiplying it by 60: \(0.5\times 60'=30'\). Therefore, \(18.5^\circ=18^\circ 30'\). Option A has 15′, which equals \(0.25^\circ\), so it is not correct. Exam tip: To convert the decimal part of a degree into minutes, always multiply it by 60.

(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.

The governing concept is coterminal angles. Two angles are coterminal when they differ by an integer multiple of 360°, because they have the same terminal side after complete rotations. To obtain a negative coterminal angle from 65°, subtract 360° once: 65° - 360° = -295°. Therefore option B is correct. Verification is immediate because -295° + 360° = 65°. Although -245°, -305°, and -325° are all negative angles, adding 360° gives 115°, 55°, and 35° respectively, not 65°. Thus they do not share the same terminal side. The essential operation is subtraction of 360°, not merely choosing any negative 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.