Angles

( -\frac{13\pi}{6}+\frac{24\pi}{6}=\frac{11\pi}{6} ), and the reference angle is ( \frac{\pi}{6} ). First bring the negative angle into (0) to (2\pi).

( \frac{4\pi}{3}+3\pi=\frac{13\pi}{3} ), and ( \frac{13\pi}{3}-4\pi=\frac{\pi}{3} ). Subtract multiples of (2\pi).

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(-75^\circ+360^\circ=285^\circ\), \(285^\circ\) is coterminal with \(-75^\circ\). The difference between \(255^\circ\) and \(-75^\circ\) is \(330^\circ\), which is not a multiple of \(360^\circ\). Exam tip: add or subtract \(360^\circ\) to check coterminal angles quickly.

To convert degrees to radians, multiply by \(\frac{\pi}{180}\): \(250^\circ=250\times\frac{\pi}{180}=\frac{25\pi}{18}\). Hence, \(\frac{x\pi}{18}=\frac{25\pi}{18}\), so \(x=25\). The nearby option 24 is incorrect because it corresponds to \(240^\circ\). Exam tip: remember \(\theta^\circ=\frac{\theta\pi}{180}\) radians.

( \frac{23\pi}{20}\times\frac{180^\circ}{\pi}=207^\circ). Use (180^\circ\div20=9^\circ) and multiply.

Sector area is ( \frac{1}{2}rs=\frac{1}{2}\times6\times9=27 ). Use this short formula when arc length is given.

The difference ( \frac{29\pi}{12}-\frac{5\pi}{12}=2\pi ), so they are coterminal. In radians, coterminal angles differ by a multiple of (2\pi).

The supplementary angle is (180^\circ-112^\circ30'=67^\circ30'). Be careful with borrowing in minute subtraction.

The complementary angle is (90^\circ-38^\circ45'=51^\circ15'). Treat (90^\circ) as (89^\circ60') while subtracting.

( \pi-\frac{25\pi}{18}=-\frac{7\pi}{18} ), and ( -\frac{7\pi}{18}+2\pi=\frac{29\pi}{18} ).

Add (2\pi) to get the positive coterminal angle. In exams, first convert a negative angle to the principal interval.

Multiply by (\frac{180}{\pi}) to convert radians into degrees. In exams, cancel (\pi) first to simplify calculation.

Subtract (3\times360^\circ) from (1180^\circ) to get (100^\circ). In exams, remove multiples of (360^\circ).

Since (765^\circ-720^\circ=45^\circ), the terminal side lies in the first quadrant. In exams, first reduce the angle between (0^\circ) and (360^\circ).

For arc length, (s=r\theta), so (\theta=\frac{s}{r}). In exams, the angle in this formula is always in radians.

(s=r\theta=5\times\frac{7\pi}{10}=\frac{7\pi}{2}) cm. In exams, put the radian angle directly in the formula.

(-940^\circ+1080^\circ=140^\circ), so the terminal side lies in the second quadrant. In exams, keep adding (360^\circ) until the angle becomes positive.

To find the principal angle, reduce the angle to the interval \([0,2\pi)\). Since \(2\pi=\frac{8\pi}{4}\), \(\frac{13\pi}{4}-2\pi=\frac{13\pi}{4}-\frac{8\pi}{4}=\frac{5\pi}{4}\). Hence, the principal angle is \(\frac{5\pi}{4}\). \(\frac{\pi}{4}\) is the reference angle, not the principal angle. Exam tip: add or subtract multiples of \(2\pi\) until the angle lies in \([0,2\pi)\).

Sector area is (A=\frac{1}{2}r^2\theta), so (24=18\theta). In exams, use (\theta) in radians in the area formula.

Multiply by (\frac{\pi}{180}) to convert degrees into radians. In exams, cancel (225) and (180) by (45).