Angles

(0<\frac{4\pi}{9}<\frac{\pi}{2}), so it lies in the first quadrant. Check the interval of small positive radian angles carefully.

( \frac{\pi}{2}<\frac{5\pi}{9}<\pi ), so it lies in the second quadrant. Remember the interval between ( \frac{\pi}{2} ) and ( \pi ).

(30'=\frac{30}{60}^\circ=0.5^\circ), so the value is (125.5^\circ). Do not write minutes directly as decimals.

(45'=\frac{45}{60}^\circ=0.75^\circ), so (76^\circ 45'=76.75^\circ). Divide minutes by (60).

(0.2^\circ\times 60'=12'), so (19.2^\circ=19^\circ 12'). Multiply the decimal part by (60).

(0.625^\circ\times 60'=37.5') and (0.5'\times 60''=30''). Convert the decimal part step by step into minutes and seconds.

(33^\circ=118800'') and (20'=1200''), so the total is (120024''). Remember (1^\circ=3600'') and (1'=60'').

Since \(1^\circ=3600''\) and \(1'=60''\), \(4585''=3600''+985''=1^\circ+985''\). Now, \(985''=16\times60''+25''=16'25''\). Therefore, \(4585''=1^\circ16'25''\). Option B incorrectly leaves \(5''\); the correct remainder is \(25''\). Exam tip: first divide seconds by \(3600\), then divide the remaining seconds by \(60\).

(2\times \frac{180^\circ}{\pi}\approx 114.6^\circ). For estimation take (1) radian as (57.3^\circ).

(3.2\times 57.3^\circ\approx 183.4^\circ). Multiply by (57.3^\circ) to convert radians approximately into degrees.

(150^\circ=\frac{150\pi}{180}=\frac{5\pi}{6}). ( \frac{2\pi}{3} ) means (120^\circ).

( \frac{7\pi}{5}\times \frac{180^\circ}{\pi}=252^\circ). If the denominator is (5), take (180^\circ\div 5=36^\circ).

(1.5\times 57.3^\circ\approx 85.95^\circ). Check the decimal carefully while choosing the nearest option.

( \theta=\frac{s}{r}=\frac{6}{9}=\frac{2}{3} ) radian. Use (s=r\theta) in arc length questions.

(s=r\theta=15\times \frac{2\pi}{5}=6\pi) cm. The formula applies directly when the angle is in radians.

(72^\circ=\frac{2\pi}{5}) and (s=20\times \frac{2\pi}{5}=8\pi) cm. Convert degrees to radians before finding the arc.

When the angle is in radians, the arc-length formula is \(s=r\theta\). Thus, \(r=\frac{s}{\theta}=\frac{11\pi}{\pi/2}=11\pi\times\frac{2}{\pi}=22\) cm. The option \(22\pi\) cm is incorrect because the \(\pi\) terms cancel. Exam tip: Use this formula directly only when the angle is measured in radians.

Area is ( \frac{1}{2}r^2\theta=\frac{1}{2}\times100\times\frac{3\pi}{10}=15\pi ). Use this formula directly when the angle is in radians.

(45^\circ=\frac{\pi}{4}) and area is ( \frac{1}{2}\times144\times\frac{\pi}{4}=18\pi ). Convert the degree angle into radians first.

(25=\frac{1}{2}\times25\times\theta), so ( \theta=2 ) radians. Isolate the unknown angle in the area formula.