Angles

Correct answer: A. (\frac{15031\pi}{21600})

(125^\circ15'30''=\frac{450930}{3600}^\circ). Multiplying by (\frac{\pi}{180}) gives (\frac{15031\pi}{21600}).

Correct answer: C. 20 seconds

Convert \(240^\circ\) into radians: \(240^\circ=\frac{240\pi}{180}=\frac{4\pi}{3}\) radians. Use \(t=\frac{\text{angular displacement}}{\text{angular speed}}\). Thus, \(t=\frac{4\pi/3}{\pi/15}=\frac{4\pi}{3}\times\frac{15}{\pi}=20\) seconds. Therefore, option C is correct. Getting \(24\) seconds indicates an incorrect degree-to-radian conversion. Exam tip: When angular speed is in radians per second, convert the angle to radians first.

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

One third of the circumference gives one third of the full angle (2\pi). Hence the central angle is (\frac{2\pi}{3}).

Correct answer: C. (120^\circ)

The difference (4\theta-120^\circ) must be a multiple of (360^\circ). For the least positive value (\theta=120^\circ).

Correct answer: C. (\frac{9\pi}{10})

In the second quadrant the angle is (180^\circ-18^\circ=162^\circ). (162^\circ=\frac{9\pi}{10}) radians.

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

Let the angle be \(x\) radians. Its complement is \(\frac{\pi}{2}-x\), and its supplement is \(\pi-x\). From the condition, \(\frac{\pi}{2}-x=\frac{1}{3}(\pi-x)\). Multiplying by 3 gives \(\frac{3\pi}{2}-3x=\pi-x\). Hence \(\frac{\pi}{2}=2x\), so \(x=\frac{\pi}{4}\). Thus, option B is correct. For the close distractor \(\frac{\pi}{3}\), the complement is \(\frac{\pi}{6}\) and the supplement is \(\frac{2\pi}{3}\); the complement is \(\frac{1}{4}\), not \(\frac{1}{3}\), of the supplement. Exam tip: use \(\frac{\pi}{2}\) for complement and \(\pi\) for supplement when working in radians.

Correct answer: B. (\frac{23\pi}{225})

(18^\circ24'=\frac{92}{5}^\circ). In radians it is (\frac{92}{5}\times\frac{\pi}{180}=\frac{23\pi}{225}).

Correct answer: D. Fourth

The angle is negative. Adding \(1440^\circ\), which represents four complete rotations, gives \(-1460^\circ+1440^\circ=-20^\circ\). The angle \(-20^\circ\) is coterminal with \(340^\circ\), and \(340^\circ\) lies between \(270^\circ\) and \(360^\circ\). Hence, its terminal side lies in the fourth quadrant. An angle in the second quadrant would lie between \(90^\circ\) and \(180^\circ\). Exam tip: For a negative angle, add a suitable multiple of \(360^\circ\) to obtain a coterminal angle between \(0^\circ\) and \(360^\circ\).

Correct answer: B. (2)

(75^\circ=\frac{5\pi}{12}) and (\frac{53\pi}{12}=4\pi+\frac{5\pi}{12}). (4\pi) contains (2) complete revolutions.

Correct answer: B. (\frac{11\pi}{12})

The time difference is (\frac{11}{2}) hours and the hour hand turns (30^\circ) per hour. The total angle is (165^\circ=\frac{11\pi}{12}).

Correct answer: B. \(1\) radian

The perimeter of a sector is \(2r+s\), where \(s\) is the arc length. From the condition, \(2r+s=3s\). Thus, \(2r=2s\), so \(s=r=8\) cm. Using \(s=r\theta\), we get \(\theta=\frac{s}{r}=\frac{8}{8}=1\) radian. At \(2\) radians, the arc length would be \(16\) cm, so the perimeter would not be three times the arc length. Exam tip: Use \(s=r\theta\) only when \(\theta\) is measured in radians.

Correct answer: C. 55°

Coterminal angles differ by an integral multiple of 360°. \(-1025^\circ+3\times360^\circ=-1025^\circ+1080^\circ=55^\circ\). Therefore, the least positive coterminal angle is 55°. Choosing 45° or 65° would not give a difference that is a whole multiple of 360°. Exam tip: For a negative angle, keep adding 360° until the result lies between 0° and 360°.

Correct answer: C. Third

Subtract a multiple of \(2\pi\) to obtain a coterminal angle: \(\frac{31\pi}{6}-4\pi=\frac{31\pi}{6}-\frac{24\pi}{6}=\frac{7\pi}{6}\). Since \(\pi<\frac{7\pi}{6}<\frac{3\pi}{2}\), the terminal side lies in the third quadrant. It is \(\frac{\pi}{6}\) past \(\pi\), so the second quadrant is not correct. Exam tip: First reduce a radian angle to a coterminal angle between \(0\) and \(2\pi\).

Correct answer: D. \(-\frac{27\pi}{8}\)

Coterminal angles differ by an integer multiple of \(2\pi\). Subtracting \(4\pi=\frac{32\pi}{8}\) from \(\frac{5\pi}{8}\) gives \(\frac{5\pi}{8}-\frac{32\pi}{8}=-\frac{27\pi}{8}\). Moreover, \(-4\pi=-\frac{32\pi}{8}< -\frac{27\pi}{8}< -\frac{16\pi}{8}=-2\pi\), so option D lies in the required interval. Option A differs from \(\frac{5\pi}{8}\) by \(-3\pi\), which is not an integer multiple of \(2\pi\). Exam tip: write coterminal angles as \(\theta+2n\pi\) and then test the required interval.

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

The point is in the second quadrant and the reference angle is (\frac{\pi}{6}). Hence the principal angle is (\frac{5\pi}{6}).

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

The fourth-quadrant ray of (y=-x) has reference angle (\frac{\pi}{4}). Hence the principal angle is (\frac{7\pi}{4}).

Correct answer: C. (4080^\circ)

One complete revolution is \\(360^\\circ\\). Therefore, 11 complete revolutions produce \\(11\\times360^\\circ=3960^\\circ\\). The additional angle is \\(2\\pi/3\\) radians. Since \\(\\pi\\) radians equals \\(180^\\circ\\), this angle equals \\(2\\times180^\\circ/3=120^\\circ\\). Adding both parts gives \\(3960^\\circ+120^\\circ=4080^\\circ\\). Hence option C is correct.

The important point is that revolutions and radians must first be expressed in the same unit before they are added. The 11 turns are not 11 degrees; each full turn contributes 360 degrees. Similarly, \\(2\\pi/3\\) is one-third of a full half-turn, or 120 degrees. The total angle is therefore 4080 degrees, not 3960 degrees, which would omit the extra rotation. The supplied answer C and explanation are accurate.

Correct answer: C. (72)

A point lies on the negative x-axis when its angle is an odd multiple of \\(\\pi\\): \\(\\pi,3\\pi,5\\pi,\\ldots\\). The condition here is \\(0<\\theta<4\\pi\\), so only \\(\\theta=\\pi\\) and \\(\\theta=3\\pi\\) are included. Substituting \\(\\theta=m\\pi/18\\), we get \\(m\\pi/18=\\pi\\), giving \\(m=18\\), and \\(m\\pi/18=3\\pi\\), giving \\(m=54\\). Their sum is \\(18+54=72\\), so option C is correct.

The strict inequalities matter: zero is excluded, and \\(4\\pi\\) itself is also excluded. Although \\(5\\pi\\) is again on the negative x-axis, it is outside the stated interval. Angles differing by \\(2\\pi\\) point in the same direction, which explains the two permitted positions, \\(\\pi\\) and \\(3\\pi\\), within this interval. Thus the supplied answer C is correct and the explanation identifies all possible values without adding an invalid endpoint.

Correct answer: C. (\frac{7\pi}{8})

A right angle is (\frac{\pi}{2}) so the reference angle is (\frac{\pi}{8}). In the second quadrant the angle is (\frac{7\pi}{8}).

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

The area ratio is (\frac{\theta}{2\pi}). From (\frac{\theta}{2\pi}=\frac{3}{4}) we get (\theta=\frac{3\pi}{2}).

Correct answer: A. (-\frac{301\pi}{720})

(-75^\circ15'=-\frac{301}{4}^\circ). In radians it is (-\frac{301}{4}\times\frac{\pi}{180}=-\frac{301\pi}{720}).

Correct answer: C. (60^\circ)

The difference is (5x+60^\circ) and it must be a multiple of (360^\circ). For the least positive value (x=60^\circ).

Correct answer: C. 6

Coterminal angles differ by \(2k\pi\), where \(k\) is an integer. Hence, \(\frac{5\pi}{12}+\frac{n\pi}{6}=\frac{17\pi}{12}+2k\pi\). Subtracting \(\frac{5\pi}{12}\) gives \(\frac{n\pi}{6}=\pi+2k\pi\), so \(n=6+12k\). The least positive value is obtained for \(k=0\), giving \(n=6\). With \(n=5\), the difference is \(\frac{5\pi}{6}\), not an integral multiple of \(2\pi\). Exam tip: test coterminality by checking whether the angle difference is an integral multiple of \(2\pi\).

Correct answer: C. after (\frac{360}{11}) minutes

The direct answer is C: \\(\frac{360}{11}\\) minutes after 3 o’clock. At exactly 3:00, the hands are 90° apart. After \\(m\\) minutes, the minute hand moves \\(6m^\circ\\), while the hour hand moves \\(0.5m^\circ\\). Thus the separation decreases at \\(6-0.5=5.5^\circ\\) per minute, so the separation is \\(90-5.5m\\). For the first later right angle, set it equal to 90° after the hands have passed through the other orientation, or equivalently use the relative movement needed to gain 90°: \\(5.5m=180^\circ\\), giving \\(m=180/5.5=360/11\\) minutes. Option C is correct, about 32 minutes 44 seconds after 3, near 3:32:44. Option A, \\(300/11\\), and B, \\(330/11\\), do not produce a 90° separation. Option D, \\(390/11\\), is later than the first occurrence. Common mistake: using only the minute-hand speed and forgetting that the hour hand also moves.

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

The arc-length formula is \(s=r\theta\), where \(\theta\) must be measured in radians. Hence, \(\frac{s}{r}=\theta\). Since \(150^\circ=150\times\frac{\pi}{180}=\frac{5\pi}{6}\), the required ratio is \(\frac{5\pi}{6}\). Note that \(\frac{2\pi}{3}\) corresponds to \(120^\circ\), not \(150^\circ\). Exam tip: In arc-length questions, convert the angle to radians first.