Angles

(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.

\(130^\circ=\frac{13\pi}{18}\), so \(k=13\). In exams, compare using \(180^\circ=\pi\).

(-\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.

(2.25=\frac{9}{4}) revolutions, so the angle is (\frac{9}{4}\times2\pi=\frac{9\pi}{2}). In exams, multiply revolutions by (2\pi).

(1) radian is approximately (57.2958^\circ), which is close to (57^\circ 18'). In exams, multiply the decimal part by (60).

The difference of coterminal angles is (2\pi n), so (\frac{\theta-\phi}{2\pi}) is an integer. In exams, check the difference, not the sum.

Since (a-b=4b) must be a multiple of (360^\circ), the least (b=90^\circ). In exams, apply the coterminal condition to the difference.

(765^\circ-720^\circ=45^\circ), which lies in the first quadrant. In exams, first find the coterminal angle and then the reference angle.

(-1000^\circ+1080^\circ=80^\circ). In exams, add multiples of (360^\circ) to a large negative angle.

The first angle gives (\frac{\pi}{4}) and the second gives (\frac{\pi}{6}), so the sum is (\frac{5\pi}{12}). In exams, reduce both angles separately.

(\frac{7\pi}{18}=70^\circ), so the original angle is (70^\circ-20^\circ=50^\circ). In exams, first convert the final measure into degrees.

(30^\circ-15'=29^\circ45'=\frac{119}{4}^\circ), so the radian measure is (\frac{119\pi}{720}). In exams, convert minutes into a fraction of a degree.