Angles

Multiply degrees by ( \frac{\pi}{180} ) to convert into radians. In exams identify multiples of (30^\circ) quickly.

(-\frac{7\pi}{6}=-210^\circ) and adding (360^\circ) gives (150^\circ). For a negative angle add (360^\circ) to get the principal angle.

Coterminal angles are obtained by adding or subtracting integral multiples of \(360^\circ\). Here, \(920^\circ-2\times360^\circ=920^\circ-720^\circ=200^\circ\). Since it is positive and less than \(360^\circ\), \(200^\circ\) is the least positive coterminal angle. \(220^\circ\) would require subtracting \(700^\circ\), which is not a multiple of \(360^\circ\). Exam tip: reduce a degree measure modulo \(360^\circ\).

\(3150'=52.5^\circ\) and \(52.5^\circ=\frac{7\pi}{24}\). Convert minutes into degrees first.

Let the angle be \(x^\circ\). Its complement is \((90-x)^\circ\). Given that the complement is \(24^\circ\) greater than the angle, \(90-x=x+24\). Thus, \(2x=66\), so \(x=33\). Therefore, the correct answer is \(33^\circ\). For example, if the angle were \(36^\circ\), its complement would be \(54^\circ\), which is only \(18^\circ\) greater. Exam tip: complementary angles always add up to \(90^\circ\).

If the angle is (x) then (180^\circ-x=3x) gives (x=45^\circ). Supplementary angles add to (180^\circ).

One complete revolution equals \(2\pi\) radians. Since \(2.75=\frac{11}{4}\), the angle is \(\frac{11}{4}\times 2\pi=\frac{11\pi}{2}\) radians. Note that \(\frac{9\pi}{2}\) represents only \(2.25\) revolutions, so it is not correct. Exam tip: multiply the number of revolutions by \(2\pi\) to convert revolutions into radians.

Since the central angle is given in radians, use the arc-length formula \(s=r\theta\). Here, \(r=14\) cm and \(\theta=\frac{3\pi}{7}\). Thus, \(s=14\times\frac{3\pi}{7}=6\pi\) cm. \(7\pi\) cm would result from incorrect simplification of the product. Exam tip: In \(s=r\theta\), the angle \(\theta\) must be in radians.

The perimeter of a sector is the sum of its two radii and its arc length: \(P=2r+r\theta\), where \(\theta\) is in radians. Here, the arc length is \(10\times\frac{2\pi}{5}=4\pi\) cm. Therefore, \(P=2(10)+4\pi=20+4\pi\) cm. Option C incorrectly takes the arc length as \(2\pi\). Exam tip: when the angle is in radians, use \(r\theta\) directly for arc length.

(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 is at (240^\circ) and the hour hand is at (80^\circ). The difference is (160^\circ).

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}).

\(\frac{19\pi}{4}-4\pi=\frac{3\pi}{4}\). Subtract multiples of \(2\pi\) to get the principal angle.