Angles

(9) revolutions per minute equals (18\pi) radians per minute. Dividing by (60) gives (\frac{3\pi}{10}).

The minute hand turns (2\pi) in (60) minutes. In (25) minutes the angle is (\frac{25}{60}\times2\pi=\frac{5\pi}{6}).

The minute hand moves through six degrees each minute, so at 40 minutes it is at \\(40\\times6=240^\\circ\\) from 12. The hour hand does not remain exactly on the 2; it moves half a degree each minute. At 2:40 its position is \\(2\\times30+40\\times0.5=80^\\circ\\). The separation is therefore \\(240^\\circ-80^\\circ=160^\\circ\\). Since this is less than \\(180^\\circ\\), it is already the smaller angle. Hence option C, \\(160^\\circ\\), is correct.

A common mistake is to place the hour hand exactly at 2 and use only \\(60^\\circ\\), which would give the wrong result. The hour hand has advanced by 20 degrees during the 40 minutes, making its actual position 80 degrees. The minute hand is at 240 degrees, and the direct difference gives 160 degrees. No wraparound adjustment is needed because the difference is below 180 degrees. Thus, the supplied calculation and answer are consistent.

The given angle is negative. Add a multiple of 360° to find a coterminal angle: \(-850^\circ+3\times360^\circ=230^\circ\). Since \(180^\circ<230^\circ<270^\circ\), its terminal side lies in the third quadrant. The fourth quadrant contains angles between \(270^\circ\) and \(360^\circ\), so it is not correct. Exam tip: Reduce any angle to a value between \(0^\circ\) and \(360^\circ\) before identifying its quadrant.

(1280^\circ-1080^\circ=200^\circ). In the third quadrant the reference angle is (200^\circ-180^\circ=20^\circ).

Coterminal angles are formed by adding integral multiples of (2\pi). Hence the form is (-\frac{3\pi}{4}+2n\pi).

In standard position, a positive angle is measured anticlockwise from the positive x-axis. Reaching the negative y-axis requires three-fourths of a full revolution: \(270^\circ=\frac{3\pi}{2}\) radians. \(\frac{\pi}{2}\) represents the positive y-axis, while \(\pi\) represents the negative x-axis. Exam tip: remember the axis angles \(0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\).

(23^\circ45'=\frac{95}{4}^\circ). In radians it is (\frac{95}{4}\times\frac{\pi}{180}=\frac{19\pi}{144}).

Multiply radians by (\frac{180}{\pi}) to convert into degrees. (\frac{17\pi}{12}\times\frac{180}{\pi}=255^\circ).

(\frac{5\pi}{3}=300^\circ) which lies in the fourth quadrant. The reference angle is (360^\circ-300^\circ=60^\circ=\frac{\pi}{3}).

Coterminal angles have the same terminal side and differ by an integer multiple of a full turn, \\(2\\pi\\). To place \\(19\\pi/4\\) in the interval from 0 to \\(2\\pi\\), subtract one full turn: \\(19\\pi/4-2\\pi=19\\pi/4-8\\pi/4=11\\pi/4\\). This is still greater than \\(2\\pi=8\\pi/4\\), so subtract another full turn: \\(11\\pi/4-8\\pi/4=3\\pi/4\\). The result lies between 0 and \\(2\\pi\\), so option C is correct.

Equivalently, subtracting \\(4\\pi\\), which is two full turns, directly gives \\(19\\pi/4-16\\pi/4=3\\pi/4\\). Adding or subtracting full turns does not change the terminal side. The answer is not \\(\\pi/4\\), because that would require a different remainder, and it is not \\(5\\pi/4\\), which is larger than the calculated remainder. Thus, the supplied answer and explanation are mathematically consistent.

The sum of supplementary angles is \(\pi\) radians. Therefore, \(x=\pi-\frac{7\pi}{18}=\frac{18\pi-7\pi}{18}=\frac{11\pi}{18}\). Hence, option C is correct. If \(\frac{13\pi}{18}\) were chosen, the sum would be \(\frac{20\pi}{18}\), not \(\pi\). Exam tip: In radians, subtract the given angle from \(\pi\) to find its supplementary angle.

In radians complementary angles add to (\frac{\pi}{2}). So the answer is (\frac{\pi}{2}-\frac{2}{5}).

For a central angle measured in radians, \(s=r\theta\). Therefore, \(\theta=\frac{s}{r}=\frac{22}{7}\) radians. \(\frac{7}{22}\) is incorrect because it reverses the required ratio. Exam tip: when arc length is given, use \(\theta=\frac{s}{r}\) directly for the angle in radians.

(1) radian (=\frac{180^\circ}{\pi}\approx57.3^\circ). In approximation questions use (\pi\approx3.14).

Using the arc-length formula \(s=r\theta\), we get \(\theta=\frac{s}{r}\). Here the arc length equals the diameter, so \(s=2r\). Therefore, \(\theta=\frac{2r}{r}=2\) radians. \(\pi\) radians represents a semicircle, whose arc length is \(\pi r\), not \(2r\). Exam tip: To find an angle in radians, use \(\theta=\frac{s}{r}\).

The distance covered in one complete revolution equals the circumference of the wheel. Its circumference is \(2\pi r=2\times\frac{22}{7}\times35=220\) cm. Since the given distance is also 220 cm, the number of revolutions is \(\frac{220}{220}=1\). In contrast, \(\frac{1}{2}\) revolution would cover only 110 cm. Exam tip: divide the total distance by the wheel's circumference to find the number of revolutions.

(54^\circ\times\frac{\pi}{180}=\frac{3\pi}{10}). Use (\frac{\pi}{180}) when converting degrees to radians.

Coterminal angles differ by an integral multiple of \(2\pi\). Since \(2\pi=\frac{12\pi}{6}\), \(\frac{13\pi}{6}-2\pi=\frac{13\pi}{6}-\frac{12\pi}{6}=\frac{\pi}{6}\). Hence, the smaller positive coterminal angle is \(\frac{\pi}{6}\). Although \(\frac{7\pi}{6}\) is positive, its difference from \(\frac{13\pi}{6}\) is \(\pi\), not a multiple of \(2\pi\). Exam tip: add or subtract \(2\pi\) to reduce an angle to \([0,2\pi)\).

In degrees coterminal angles are obtained by adding multiples of (360^\circ). So the form is (110^\circ+360^\circ k).

(\frac{\pi}{9}\times\frac{180^\circ}{\pi}=20^\circ). Multiply by (\frac{180}{\pi}) to convert radians into degrees.

(72^\circ\times\frac{\pi}{180}=\frac{2\pi}{5}). Divide (72) and (180) by (36) while simplifying.

Put (\frac{1}{2}r^2\theta=45\pi). From (\frac{81}{2}\theta=45\pi) we get (\theta=\frac{10\pi}{9}).

It turns (2\pi) radians in (60) minutes. In (7.5) minutes it turns (\frac{7.5}{60}\times2\pi=\frac{\pi}{4}).

The direct answer is A, \\(\frac{\pi}{6}\\). Coterminal radian angles differ by multiples of \\(2\pi\\). Start with \\(-\frac{23\pi}{6}\\). Since \\(2\pi=\frac{12\pi}{6}\\), add suitable full turns until the result lies between 0 and \\(2\pi\\): \\(-\frac{23\pi}{6}+\frac{24\pi}{6}=\frac{\pi}{6}\\), which is in the required interval. Option A is correct. Option B, \\(\frac{\pi}{3}\\), differs from the original by a non-integer number of full turns. Option C, \\(\frac{5\pi}{6}\\), is likewise not coterminal. Option D, \\(\frac{11\pi}{6}\\), is positive and in the interval, but it is not obtained by adding an integer multiple of \\(2\pi\\) to the given angle. The supplied calculation using \\(4\pi\\) is exactly the same: \\(-\frac{23\pi}{6}+4\pi=\frac{\pi}{6}\\). Memory cue: in radians, add or subtract \\(2\pi\\), not \\(360\\).