Angles

The resulting angle is (125^\circ-410^\circ=-285^\circ), which is (-\frac{19\pi}{12}) radians. In exams, take anticlockwise as positive and clockwise as negative.

Coterminal angles differ by an integer multiple of \(2\pi\). Since \(2\pi=\frac{14\pi}{7}\), subtracting \(4\pi=\frac{28\pi}{7}\) gives \(\frac{31\pi}{7}-\frac{28\pi}{7}=\frac{3\pi}{7}\). This is positive and is the least positive coterminal angle. \(\frac{17\pi}{7}\) is also coterminal, but subtracting \(2\pi\) from it gives \(\frac{3\pi}{7}\); \(\frac{10\pi}{7}\) is not coterminal. Exam tip: reduce an angle to \([0,2\pi)\) by adding or subtracting suitable multiples of \(2\pi\).

For a non-zero angle, the numerical ratio is always (180/\pi), not (2). In exams, identify the numerical ratio between degree and radian measures.

(\frac{\pi}{3}=60^\circ), so the new angle is (45^\circ=\frac{\pi}{4}). In exams, convert mixed units into one unit.

The hour hand is at (220^\circ) and the minute hand at (120^\circ), so the difference is (100^\circ). In exams, do not forget the extra movement of the hour hand.

The hour hand is at (85^\circ) and the minute hand at (300^\circ), so the smaller angle is (145^\circ). In exams, subtract the larger difference from (360^\circ).

Complementary angles sum to (90^\circ), so the larger angle is (54^\circ=\frac{3\pi}{10}). In exams, first find the total parts of the ratio.

Supplementary angles sum to (180^\circ), so the smaller angle is (75^\circ=\frac{5\pi}{12}). In exams, divide (180^\circ) in the given ratio.

(\frac{7\pi}{20}\times\frac{180^\circ}{\pi}=63^\circ), so (x=63). In exams, cancel (\pi) and simplify.

(\frac{13\pi}{15}=156^\circ), which is between (90^\circ) and (180^\circ). In exams, converting to degrees often makes quadrant identification easier.

The principal angle of (-\frac{23\pi}{8}) is (\frac{9\pi}{8}), which lies in the third quadrant. In exams, add (2\pi) to a negative angle.

The positive principal angle of (845^\circ) is (125^\circ), so the negative coterminal angle is (125^\circ-360^\circ=-235^\circ). In exams, subtract (360^\circ) for the negative answer.

(\frac{19\pi}{6}=2\pi+\frac{7\pi}{6}), so one complete revolution and (\frac{7\pi}{6}) remain. In exams, treat (2\pi) as one revolution.

The minute hand moves clockwise, so the angle is (-\frac{47}{60}\times2\pi=-\frac{47\pi}{30}). In exams, assign the sign according to direction.

(5) hours (20) minutes (=\frac{16}{3}) hours, which is (\frac{4}{9}) of (12) hours. The angle is (\frac{4}{9}\times2\pi=\frac{8\pi}{9}).

(2A=40^\circ 30') and (40^\circ 30'-12^\circ 45'=27^\circ 45'). In exams, remember borrowing while subtracting minutes.

(27^\circ 45'=\frac{111}{4}^\circ), so the radian measure is (\frac{111\pi}{720}=\frac{37\pi}{240}). In exams, write minutes as (\frac{45}{60}^\circ).

(\frac{11\pi}{9}\times\frac{180^\circ}{\pi}=220^\circ), so (x=220). In exams, apply degree conversion directly.

(\frac{35\pi}{3}\div2\pi=\frac{35}{6}), which is not an integer. In exams, check (2\pi n) for coterminal angles.

Adding (2\pi) three times to (-\frac{41\pi}{10}) gives (\frac{19\pi}{10}). In exams, keep adding (2\pi) until the angle enters the interval.

(75^\circ+360^\circ=435^\circ=\frac{29\pi}{12}), which lies between (2\pi) and (3\pi). In exams, add revolutions according to the given interval.

The remainder of (540^\circ) is (180^\circ), so (180^\circ+\theta\equiv70^\circ) and (\theta=250^\circ). In exams, reduce the fixed part first.

(-\frac{5\pi}{3}+2\pi=\frac{\pi}{3}). In exams, add (2\pi) to convert a negative angle into a positive principal angle.

(\theta=\frac{11}{21}) radians and the degree measure is (\frac{11}{21}\times\frac{180^\circ\times7}{22}=30^\circ). In exams, convert radians to degrees carefully.

(72^\circ=\frac{2\pi}{5}) and (r=\frac{s}{\theta}=\frac{4\pi}{2\pi/5}=10) cm. In exams, keep the angle in radians when using the formula.