Angles

The angle is (\frac{3\pi}{8}\times10=\frac{15\pi}{4}) radians, which is (675^\circ). In exams, first find the total radians.

\(\frac{11\pi}{36}\times\frac{180^\circ}{\pi}=55^\circ\). In exams, cancel \(\pi\) first.

(81^\circ=\frac{9\pi}{20}), so (\frac{3x\pi}{20}=\frac{9\pi}{20}) gives (x=3). In exams, write both angles in the same unit.

(-420^\circ30'+720^\circ=299^\circ30'). In exams, add multiples of (360^\circ) even for mixed degree-minute angles.

(\frac{58\pi}{9}-6\pi=\frac{4\pi}{9}). In exams, subtract multiples of (2\pi) to keep the angle in the interval.

The difference between the two angles is (4\pi), which is a multiple of (2\pi). Therefore, they are coterminal.

The principal angle in the fourth quadrant is (333^\circ), and its greatest negative coterminal angle is (333^\circ-360^\circ=-27^\circ). In exams, keep the greatest negative angle above (-360^\circ).

(36+18\theta=48), so (\theta=\frac{2}{3}) radians. In degrees, this is (\frac{2}{3}\times\frac{180}{\pi}=\frac{120}{\pi}^\circ).

(218^\circ24'=218.4^\circ=\frac{1092}{5}^\circ), so the radian measure is (\frac{91\pi}{75}). In exams, convert (24') into (0.4^\circ).

(54^\circ=\frac{3\pi}{10}) and from (s=r\theta), (r=\frac{9\pi}{\frac{3\pi}{10}}=30). In exams, convert degrees into radians before using the arc formula.

In a sector, (A=\frac{1}{2}rs), and (A=3s) is given. Hence (\frac{1}{2}r=3), giving (r=6).

The complementary angle is (\frac{\pi}{2}-\frac{2\pi}{7}=\frac{3\pi}{14}). In degrees, it is (\frac{270}{7}^\circ).

(-\frac{101\pi}{18}+6\pi=\frac{7\pi}{18}). In exams, writing (6\pi) as (\frac{108\pi}{18}) makes addition easy.

(210^\circ=\frac{7\pi}{6}), while (\frac{5\pi}{6}=150^\circ). In exams, convert all options into the same unit before comparing.

(\theta=1.65\pi), which lies between (\frac{3\pi}{2}) and (2\pi). Therefore, the angle is in the fourth quadrant.

The rotation is (-\frac{11}{8}\times2\pi=-\frac{11\pi}{4}), and adding (4\pi) gives (\frac{5\pi}{4}). In exams, first convert revolutions into radians.

The reflex angle is (360^\circ-126^\circ40'=233^\circ20'). In exams, subtract using (1^\circ=60').

The angle is (72^\circ-5\times360^\circ=-1728^\circ). In exams, going behind means subtracting multiples of (360^\circ).

A straight angle is (\pi) radians, so the remaining angle is (\pi-3) radians. In degrees, it is (\frac{180(\pi-3)}{\pi}).

From (s=r\theta), (\theta=\frac{\frac{5\pi r}{6}}{r}=\frac{5\pi}{6}), which is (150^\circ). In exams, cancel (r) to find the angle.