Angles

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.

Sector area is (\frac{1}{2}r^2\theta). So (\frac{1}{2}\times36\times\frac{5\pi}{9}=10\pi).

(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.

\(\theta=\frac{s}{r}=\frac{8\pi}{16}=\frac{\pi}{2}\). Apply the arc length formula in reverse.

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

The hour hand turns (30^\circ) in (1) hour. In (\frac{10}{3}) hours the angle is (100^\circ).

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

\(-\frac{23\pi}{6}+4\pi=\frac{\pi}{6}\). Add multiples of \(2\pi\) to a negative radian angle.

(\frac{11\pi}{10}) lies in the third quadrant. The reference angle is (\frac{11\pi}{10}-\pi=\frac{\pi}{10}).

Coterminal angles have the same initial and terminal arms, and their measures differ by a whole multiple of \(360^\circ\). Here, \(495^\circ-135^\circ=360^\circ\), so A is correct. In B, the difference is only \(90^\circ\). Exam tip: check the difference first.

(12^\circ30'=12.5^\circ=\frac{25}{2}^\circ). In radians it is (\frac{25}{2}\times\frac{\pi}{180}=\frac{5\pi}{72}).