Angles

Correct answer: A. 13π/12

Direct answer: A, 13π/12. Coterminal angles have the same terminal side, so their difference is an integer multiple of one full revolution, 2π. The requested answer must lie strictly between 0 and 2π. Express one revolution with denominator 12: 2π = 24π/12. Subtract one revolution from the given angle: 37π/12 − 24π/12 = 13π/12. This is positive and smaller than 24π/12, so it is in the required interval. A is correct. B, 11π/12, is not obtained by subtracting an integer multiple of 2π. C, 7π/12, and D, π/12, likewise have different remainders and are not coterminal with the given angle. Memory cue: reduce an angle by complete turns, not by arbitrary fractions of a turn.

Correct answer: B. ( \frac{19\pi}{10} )

(-\frac{41\pi}{10}+\frac{60\pi}{10}=\frac{19\pi}{10}). Add enough multiples of (2\pi) to a negative angle.

Correct answer: C. 16π/9

Direct answer: C, 16π/9. Coterminal angles differ by an integer multiple of one complete revolution, 2π. The usual principal-angle interval here is [0,2π). Express 2π with denominator 9: 2π=18π/9. Subtract two complete revolutions from 52π/9: 52π/9−18π/9−18π/9=16π/9. This value is non-negative and less than 18π/9, so it lies in the required principal interval. A, 14π/9, is not obtained by subtracting an integral multiple of 18π/9 from 52π/9. B, 8π/9, and D, 2π/9, fail the same coterminality test. The intermediate result after one subtraction is 34π/9, but that is still larger than 2π, so stopping there is a mistake. Memory cue: reduce the coefficient of π modulo 18 when the denominator is 9.

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

The answer is option A, pi/5. Angles that differ by a full turn, 2pi radians, have the same position. The given angle is -29pi/5. Add 2pi=10pi/5 repeatedly until the result lies between 0 and 2pi. First, -29pi/5+10pi/5=-19pi/5; second, -19pi/5+10pi/5=-9pi/5; third, -9pi/5+10pi/5=pi/5. This lies between 0 and 2pi, so it is the required coterminal angle. Option A, pi/5, is correct. Option B, 3pi/5, would differ from the given angle by 32pi/5, which is not an integer multiple of 2pi. Option C, 7pi/5, and option D, 9pi/5, also do not result from adding whole turns to the given angle. A shorter calculation is -29pi/5+30pi/5=pi/5. Always check that the final answer is in the interval 0 to 2pi.

Correct answer: B. \(50^\circ\)

To obtain the principal coterminal angle, subtract a multiple of \(360^\circ\) so that the result lies between \(0^\circ\) and \(360^\circ\). \(2210^\circ-6\times360^\circ=2210^\circ-2160^\circ=50^\circ\). Hence, \(50^\circ\) is correct. \(40^\circ\) is not coterminal with \(2210^\circ\), since their difference is not an integral multiple of \(360^\circ\). Exam tip: divide the angle by \(360^\circ\) and use the remainder.

Correct answer: C. (305^\circ)

(-1855^\circ+2160^\circ=305^\circ). Add a suitable multiple of (360^\circ) to a large negative angle.

Correct answer: D. They are coterminal angles

(1485^\circ-45^\circ=1440^\circ=4\times360^\circ), so they are coterminal. If the difference is a multiple of (360^\circ), consider them coterminal.

Correct answer: A. First quadrant

(-640^\circ+720^\circ=80^\circ), and (80^\circ) lies in the first quadrant. First convert a negative angle into a positive coterminal angle.

Correct answer: D. Fourth quadrant

( \frac{3\pi}{2}<\frac{23\pi}{14}<2\pi ), so it lies in the fourth quadrant. Compare radian limits with a common denominator.

Correct answer: B. Second quadrant

( \frac{\pi}{2}<\frac{6\pi}{11}<\pi ), so the terminal side lies in the second quadrant. The interval between ( \frac{\pi}{2} ) and ( \pi ) is the second quadrant.

Correct answer: C. Third quadrant

( \pi<\frac{15\pi}{11}<\frac{3\pi}{2} ), so it lies in the third quadrant. After ( \pi ) and before ( \frac{3\pi}{2} ) is the third quadrant.

Correct answer: C. Third quadrant

( -\frac{5\pi}{8}+2\pi=\frac{11\pi}{8} ), and it lies in the third quadrant. Add (2\pi) to a negative radian angle to check its position.

Correct answer: B. (72.5^\circ)

The direct answer is option B, \(72.5^\circ\). In degree-minute notation, one degree contains 60 minutes, so 30 minutes is half a degree: \(30'=\frac{30}{60}^\circ=0.5^\circ\). Add this to the whole-degree part: \(72^\circ+0.5^\circ=72.5^\circ\). Option B is correct. Option A, \(72.3^\circ\), wrongly treats 30 minutes as 0.3 degree. Option C, \(73.0^\circ\), treats 30 minutes as a full degree. Option D, \(72.25^\circ\), would represent 15 minutes, because \(15/60=0.25\), not 30 minutes. The key rule is that minutes are a fraction of a degree, not decimal digits written after the degree number. Memory cue: divide arc-minutes by 60 before adding.

Correct answer: A. (41^\circ 7'30'')

(0.125^\circ\times60'=7.5'), and (0.5'\times60''=30''). Convert the decimal part step by step into minutes and seconds.

Correct answer: A. (98316'')

(27^\circ=97200''), and (18'=1080''), so the total is (98316''). Remember (1^\circ=3600'') and (1'=60'').

Correct answer: B. \(2^\circ 3'15''\)

Since \(1^\circ=3600''\), \(7395''=2\times3600''+195''\). Now, \(195''=3\times60''+15''=3'15''\). Therefore, \(7395''=2^\circ3'15''\), so option B is correct. Option C has an incorrect conversion of the remaining seconds into minutes and seconds. Exam tip: first divide total seconds by 3600 for degrees, then divide the remainder by 60 for minutes.

Correct answer: C. \(160.4^\circ\)

The direct answer is option C, \(160.4^\circ\). A radian measures an angle by comparing arc length with radius. Since a full circle is \(2\pi\) radians and \(360^\circ\), one radian equals \(180/\pi\) degrees, approximately \(57.3^\circ\). Therefore, \(2.8\) radians equals \(2.8\times180/\pi\), or approximately \(2.8\times57.3=160.44^\circ\). The nearest listed value is \(160.4^\circ\). Option A, \(150.4^\circ\), is too small and would result from using an incorrect conversion. Option B, \(155.8^\circ\), is also below the calculated value. Option C is correct because it is closest to \(160.44^\circ\). Option D, \(168.2^\circ\), is too large. Remember: to change radians to degrees, multiply by \(180/\pi\), or approximately by \(57.3\). Do not multiply by \(\pi/180\), because that converts degrees to radians.

Correct answer: B. (43.0^\circ)

(0.75\times57.3^\circ\approx43.0^\circ). The same conversion factor applies for small radian values too.

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

( \theta=\frac{s}{r}=\frac{14\pi}{21}=\frac{2\pi}{3} ). Use ( \theta=\frac{s}{r} ) to find the angle from arc length.

Correct answer: C. (10\pi) cm

(s=r\theta=16\times\frac{5\pi}{8}=10\pi) cm. When the angle is in radians, (s=r\theta) applies directly.

Correct answer: B. (10\pi) cm

(75^\circ=\frac{5\pi}{12}), and (s=24\times\frac{5\pi}{12}=10\pi) cm. Convert degrees to radians before finding the arc.

Correct answer: C. 24 cm

When the angle is in radians, the arc-length formula is \(s=r\theta\). Therefore, \(r=\frac{s}{\theta}=\frac{18}{3/4}=18\times\frac{4}{3}=24\) cm. Hence, option C is correct. Option D may result from multiplying \(18\) by \(\frac{3}{4}\), but finding the radius requires division by \(\theta\). Exam tip: In arc-length questions, first check that the angle is expressed in radians.

Correct answer: C. (70\pi) square cm

Area is ( \frac{1}{2}r^2\theta=\frac{1}{2}\times196\times\frac{5\pi}{7}=70\pi ). Use this formula directly with a radian angle.

Correct answer: B. (27\pi) square cm

(120^\circ=\frac{2\pi}{3}), and area is ( \frac{1}{2}\times81\times\frac{2\pi}{3}=27\pi ). Convert the degree angle into radians first.

Correct answer: B. (3) radians

From (54=\frac{1}{2}\times36\times\theta), ( \theta=3 ) radians. Isolate the unknown angle in the sector area formula.