Angles

(2.5\times 57.3^\circ\approx 143.25^\circ). Estimate by taking (1) radian as (57.3^\circ).

\(135^\circ\) lies between \(90^\circ\) and \(180^\circ\). This is the interval of the second quadrant.

The terminal side lies on the negative (x)-axis at (180^\circ). Axis angles are not counted in quadrants.

(540^\circ-360^\circ=180^\circ) so the terminal side lies on the negative (x)-axis. First find the coterminal angle.

( -90^\circ ) reaches the negative (y)-axis by clockwise rotation. Watch the negative direction carefully.

(30'=\frac{30}{60}^\circ=0.5^\circ) so the total is (18.5^\circ). Divide minutes by (60) to convert them into degrees.

(15'=\frac{15}{60}^\circ=0.25^\circ) so (42^\circ 15'=42.25^\circ). Do not write minutes directly as decimals.

(0.75^\circ\times 60'=45') so (23.75^\circ=23^\circ 45'). Multiply the decimal part by (60).

Since \(1^\circ=3600''\) and \(1'=60''\), \(12^\circ 20' 30''=12\times3600+20\times60+30=43200+1200+30=44430''\). Therefore, option A is correct. \(43320''\) results from an incorrect conversion of the minutes and seconds. Exam tip: when converting an angle to seconds, multiply degrees by \(3600\) and minutes by \(60\).

Since \(1^\circ=3600''\), \(9050''=2\times3600''+1850''\), giving \(2^\circ\). Next, \(1850''=30\times60''+50''\), so the remaining part is \(30'\ 50''\). Therefore, \(9050''=2^\circ 30' 50''\). Option B does not split the remaining seconds correctly into minutes and seconds. Exam tip: divide total seconds by \(3600\) first for degrees, then divide the remainder by \(60\) for minutes.

In radians ( \theta=\frac{s}{r}=\frac{14}{7}=2 ). Use (s=r\theta) in arc length questions.

(s=r\theta=5\times \frac{3\pi}{5}=3\pi) cm. This formula applies directly when the angle is in radians.

( \theta=\frac{s}{r}=\frac{6\pi}{12}=\frac{\pi}{2} ) radians. Rearrange (s=r\theta) as ( \theta=\frac{s}{r} ).

(60^\circ=\frac{\pi}{3}) and (s=10\times \frac{\pi}{3}=\frac{10\pi}{3}) cm. Convert the angle into radians before finding arc length.

(135^\circ=\frac{3\pi}{4}) and (s=8\times \frac{3\pi}{4}=6\pi) cm. It is necessary to convert a degree angle into radians first.

The arc-length formula is \(s=r\theta\), where \(\theta\) must be in radians. Thus, \(r=\frac{s}{\theta}=\frac{9}{3}=3\) cm. Therefore, the correct answer is 3 cm. The option 27 cm results from multiplying \(s\) by \(\theta\), whereas finding the radius requires division. Exam tip: For arc-length questions, use \(s=r\theta\) only when the angle is in radians.

( \theta=\frac{s}{r}=\frac{18}{12}=\frac{3}{2} ) radians. The arc length and radius must have the same unit.

For an angle measured in radians, \(\theta=\frac{s}{r}\), where \(s\) is the arc length and \(r\) is the radius. Here \(s=r\), so \(\theta=\frac{r}{r}=1\) radian. \(\pi\) radians represents a semicircle, so it is not correct here. Exam tip: For arc-length questions, first use \(\theta=\frac{s}{r}\).

(s=r\theta=14\times \frac{\pi}{7}=2\pi) cm. Apply (s=r\theta) directly with a radian angle.

The sector area is ( \frac{1}{2}r^2\theta ) so ( \frac{1}{2}\times36\times\frac{\pi}{3}=6\pi ). Apply the formula directly when the angle is in radians.