Angles

(75^\circ=\frac{75\pi}{180}=\frac{5\pi}{12}). Multiply by ( \frac{\pi}{180} ) to convert degrees to radians.

(105^\circ=\frac{105\pi}{180}=\frac{7\pi}{12}). Simplify the fraction before choosing the answer.

(165^\circ=\frac{165\pi}{180}=\frac{11\pi}{12}). Multiples of (15^\circ) can lead to denominator (12).

Direct answer: Option D, \(\frac{13\pi}{12}\) radians. The conversion rule is degrees multiplied by \(\frac{\pi}{180}\), because 180 degrees equals \(\pi\) radians. Therefore, \(195^\circ=195\times\frac{\pi}{180}=\frac{195\pi}{180}\). Divide numerator and denominator by 15: \(\frac{195}{180}=\frac{13}{12}\), so the result is \(\frac{13\pi}{12}\) radians. Option A, \(\frac{11\pi}{12}\), represents 165 degrees, not 195 degrees. Option B, \(\frac{7\pi}{6}\), represents 210 degrees. Option C, \(\frac{4\pi}{3}\), represents 240 degrees. Option D is correct because its degree conversion returns to 195 degrees: \(\frac{13\pi}{12}×\frac{180}{\pi}=195^\circ\). Memory cue: divide the degree number by 180 and attach \(\pi\), then simplify.

(255^\circ=\frac{255\pi}{180}=\frac{17\pi}{12}). Dividing by (15) is useful for simplification.

(285^\circ=\frac{285\pi}{180}=\frac{19\pi}{12}). Counting in ( \frac{\pi}{12} ) steps helps for such angles.

To convert degrees to radians, multiply the angle by \(\frac{\pi}{180}\): \((-135^\circ)\times\frac{\pi}{180}=-\frac{135\pi}{180}=-\frac{3\pi}{4}\). Hence, the correct answer is \(-\frac{3\pi}{4}\). Note that \(-\frac{\pi}{4}\) corresponds to \(-45^\circ\), not \(-135^\circ\). Exam tip: Always retain the negative sign when converting a negative angle to radians.

( \frac{5\pi}{12}\times \frac{180^\circ}{\pi}=75^\circ). Cancel ( \pi ) and divide (180^\circ) by the denominator.

( \frac{7\pi}{12}=\frac{7\times 180^\circ}{12}=105^\circ). Use ( \frac{180^\circ}{\pi} ) for radians to degrees.

( \frac{13\pi}{12}\times \frac{180^\circ}{\pi}=195^\circ). Remember ( \frac{\pi}{12}=15^\circ).

( \frac{17\pi}{12}=17\times 15^\circ=255^\circ). Take ( \frac{\pi}{12}=15^\circ) for a quick solution.

( \frac{19\pi}{12}\times \frac{180^\circ}{\pi}=285^\circ). If the denominator is (12) multiply the numerator by (15^\circ).

The answer is option C, 345 degrees. Use pi radians=180 degrees. In 23pi/12, each pi corresponds to 180 degrees, so the value is (23/12) times 180 degrees. Since 180/12=15, this becomes 23 times 15=345 degrees. Option C is therefore correct. Option A, 335 degrees, is 10 degrees too small and would correspond to a different radian value. Option B, 340 degrees, is also not the result of multiplying 23 by 15. Option D, 355 degrees, is 10 degrees too large. A useful check is that 23pi/12 is one pi/12 less than 2pi, because 2pi=24pi/12. One pi/12 equals 15 degrees, so 360-15=345 degrees. Be careful not to round a value close to 360 degrees to 360; exact conversion is required.

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(840^\circ-2\times360^\circ=120^\circ\), the coterminal angle lying between \(0^\circ\) and \(360^\circ\) is \(120^\circ\). Although \(90^\circ\) may seem close, \(840^\circ-90^\circ=750^\circ\), which is not a multiple of \(360^\circ\). Exam tip: add or subtract multiples of \(360^\circ\) to bring an angle into the required interval.

Coterminal angles differ by an integral multiple of \(360^\circ\). Adding \(3\times360^\circ=1080^\circ\) to \(-780^\circ\) gives \(300^\circ\): \(-780^\circ+1080^\circ=300^\circ\). Hence, \(300^\circ\) is correct. \(240^\circ\) is not coterminal because \(240^\circ-(-780^\circ)=1020^\circ\), which is not a multiple of \(360^\circ\). Exam tip: Keep adding \(360^\circ\) to a negative angle until the least positive angle is obtained.

( -1020^\circ+1080^\circ=60^\circ ). Add a suitable multiple of (360^\circ) for large negative angles.

( \frac{25\pi}{6}-\frac{24\pi}{6}=\frac{\pi}{6} ). Subtract (2\pi) in radians to find the principal angle.

( -\frac{17\pi}{4}+\frac{24\pi}{4}=\frac{7\pi}{4} ). Add multiples of (2\pi) to a negative radian angle.

( -\frac{11\pi}{3}+\frac{12\pi}{3}=\frac{\pi}{3} ). Keeping the same denominator makes adding (2\pi) easy.

(725^\circ-5^\circ=720^\circ=2\times 360^\circ) so they are coterminal. If the difference is a multiple of (360^\circ) the angles are coterminal.

( -220^\circ+360^\circ=140^\circ ) and (140^\circ) lies in the second quadrant. First convert a negative angle to a positive coterminal angle.

( -310^\circ+360^\circ=50^\circ ) and (50^\circ) lies in the first quadrant. The coterminal angle gives the quadrant easily.

( \frac{8\pi}{5}=288^\circ ) so it lies in the fourth quadrant. Convert radians to degrees when needed.

( \frac{7\pi}{10}=126^\circ ) and it lies between (90^\circ) and (180^\circ). Check the interval to decide the quadrant.

( \frac{11\pi}{10}=198^\circ ) so it is in the third quadrant. The interval between (180^\circ) and (270^\circ) is the third quadrant.