Angles

Correct answer: B. (286^\circ21'49'')

(5) radians (=5\times\frac{180^\circ}{\pi}). Using (\pi=\frac{22}{7}) gives approximately (286^\circ21'49'').

Correct answer: D. No because (\frac{1980}{360}) is not an integer

The difference of coterminal angles is an integer multiple of (360^\circ). (\frac{1980}{360}=\frac{11}{2}) is not an integer.

Correct answer: B. (\frac{2}{5})

The perimeter is (24+12\theta) and the area is (72\theta). From (24+12\theta=72\theta) we get (\theta=\frac{2}{5}).

Correct answer: A. (\frac{\pi}{4})

The difference (\frac{5\pi}{2}-2\theta) must be a multiple of (2\pi). In the given interval (\theta=\frac{\pi}{4}) is suitable.

Correct answer: D. \(\frac{660}{\pi}\)

For arc length, \(s=r\theta\), where \(\theta\) must be measured in radians. Here \(s=550\) and \(\theta=\frac{5\pi}{6}\). Hence, \(r=\frac{s}{\theta}=\frac{550}{5\pi/6}=\frac{550\times6}{5\pi}=\frac{660}{\pi}\) m. Therefore, option D is correct. The value \(\frac{630}{\pi}\) may result from an error while multiplying or simplifying the fractions. Exam tip: Apply \(s=r\theta\) directly only when the angle is in radians.

Correct answer: A. No non-zero angle is possible

Direct answer: Option A, no non-zero angle is possible; the only angle is 0 radians. Let the radian measure be x. The conversion rule is degree measure = x(180/pi). The condition says x(180/pi) = 60x. Moving terms gives x(180/pi - 60) = 0. Since 180/pi is about 57.3 and is not 60, the bracket is not zero, so x = 0. Option A is correct. Option B, pi/6, would be 30 degrees, but 60 times pi/6 is about 31.4, not 30. Option C, pi/3, is 60 degrees, but 60(pi/3) is about 62.8. Option D, pi/2, is 90 degrees, not 60(pi/2). Memory cue: always convert radians using 180/pi before comparing.

Correct answer: B. (40^\circ) and (\frac{11\pi}{9})

(220^\circ) is in the third quadrant so the reference angle is (40^\circ). (220^\circ=\frac{11\pi}{9}) radians.

Correct answer: D. \(\frac{29\pi}{18}\)

Coterminal angles differ by an integral multiple of \(2\pi\). Add \(2\pi\) to the given negative angle: \(-\frac{7\pi}{18}+2\pi=-\frac{7\pi}{18}+\frac{36\pi}{18}=\frac{29\pi}{18}\). This value lies between \(0\) and \(2\pi\), so it is the principal positive angle. The difference between \(\frac{25\pi}{18}\) and \(-\frac{7\pi}{18}\) is \(\frac{16\pi}{9}\), which is not an integral multiple of \(2\pi\). Exam tip: to convert a negative angle into its principal positive form, add \(2\pi\) as needed.

Correct answer: C. (12)

The interior angle of a regular polygon is (\frac{(n-2)\pi}{n}). From (\frac{(n-2)\pi}{n}=\frac{5\pi}{6}) we get (n=12).

Correct answer: B. (\frac{7\pi}{6})

The net angle is (\frac{17\pi}{6}-\frac{5\pi}{3}=\frac{7\pi}{6}). Take counterclockwise as positive and clockwise as negative.

Correct answer: D. (\frac{4\pi}{3})

The clockwise angle is (-\frac{14\pi}{3}). Adding (6\pi) gives the principal angle (\frac{4\pi}{3}).

Correct answer: C. (\frac{2\pi}{3})

For equal radii sector area is proportional to angle. The larger angle is (75^\circ\times\frac{8}{5}=120^\circ=\frac{2\pi}{3}).

Correct answer: B. (5\pi/8)

(112^\circ 30') means (112.5^\circ), so the value is (5\pi/8). In exams, first convert minutes into degrees.

Correct answer: D. (355^\circ)

Adding (360^\circ) three times to (-725^\circ) gives (355^\circ). In exams, keep the answer between (0^\circ) and (360^\circ).

Correct answer: A. (\frac{11\pi}{3})

One complete revolution is (2\pi) radians, so the measure is (\frac{11}{6}\times2\pi=\frac{11\pi}{3}). In exams, take a full turn as (2\pi).

Correct answer: C. (-170^\circ)

Multiplying (-\frac{17\pi}{18}) by (180^\circ/\pi) gives (-170^\circ). The negative sign shows direction.

Correct answer: B. Second quadrant

The coterminal angle of (1234^\circ) is (154^\circ), which lies in the second quadrant. In exams, first find the remainder after division by (360^\circ).

Correct answer: A. (\frac{\pi}{6})

Adding (2\pi) five times to (-\frac{29\pi}{6}) gives (\frac{\pi}{6}). Always keep the principal angle in the given interval.

Correct answer: D. (\frac{5}{2})

Using (s=r\theta), (\theta=\frac{35}{14}=\frac{5}{2}) radians. In exams, divide arc length by radius.

Correct answer: C. (6\pi) cm

(135^\circ=\frac{3\pi}{4}) and (s=8\times\frac{3\pi}{4}=6\pi). In exams, first convert degrees into radians.

Correct answer: B. (-\frac{7\pi}{4})

Coterminal angles must differ by an integral multiple of (2\pi). The difference for (-\frac{7\pi}{4}) does not satisfy this.

Correct answer: D. (\frac{\pi}{180})

Since (180^\circ=\pi) radians, (1^\circ=\frac{\pi}{180}) radians. Remember this basic conversion for exams.

Correct answer: A. (114\frac{6}{11}^\circ)

(2\times\frac{180^\circ}{\pi}=2\times\frac{180^\circ\times7}{22}=114\frac{6}{11}^\circ). In exams, use the given value of (\pi).

Correct answer: C. (\frac{5\pi}{6})

(\theta=150^\circ), and (150^\circ=\frac{5\pi}{6}). In exams, first find the angle, then convert into radians.

Correct answer: B. (\frac{\pi}{2})

A straight angle is (180^\circ), so (\frac{1}{5}) is (36^\circ) and the total is (90^\circ). In exams, translate words step by step.