Angles

(1) radian is ( \frac{180^\circ}{\pi} \approx 57.3^\circ ). Use ( \pi\approx 3.14 ) for approximation.

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.

To convert the fractional part of degrees to minutes, use \(1^\circ=60'\).
\[23.75^\circ = 23^\circ + 0.75^\circ\]
Now convert \(0.75^\circ\) to minutes:
\[0.75\times 60'=45'\]
So \(23.75^\circ=23^\circ\ 45'\).

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.

The answer is option C, 3pi cm. For an angle measured in radians, arc length is s=rtheta, where r is the radius and theta is the angle in radians. Here r=5 cm and theta=3pi/5. Substitution gives s=5(3pi/5) cm. The 5 cancels with the denominator 5, leaving s=3pi cm. Option A, 2pi cm, would result from using a different angle or an incorrect multiplication. Option B, 5pi cm, fails to cancel the denominator and is too large. Option C, 3pi cm, follows exactly from the formula. Option D, 3pi/5 cm, is only the angle expression and does not include multiplication by the radius. Thus C is correct. The important condition is that theta must be in radians. If the angle were in degrees, it would first need conversion before using s=rtheta.

( \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 \(\theta=\frac{s}{r}\).

For a central angle measured in radians, the arc-length formula is s = rθ. This relationship follows from the fact that an angle of one radian subtends an arc whose length equals the radius. Here r = 14 cm and θ = π/7, so s = 14 × π/7 cm = 2π cm. Therefore option B is correct. No conversion into degrees is needed because the angle is already expressed in radians. Option A misses a factor of 2. Option C results from failing to cancel the factor 14 with the denominator 7 correctly, while option D treats the angle as though it were exactly 1 radian. The exact answer is 2π cm, approximately 6.28 cm.

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.

(37.5^\circ=\frac{37.5\pi}{180}=\frac{5\pi}{24}). You can also treat (37.5^\circ) as ( \frac{75^\circ}{2} ).

Coterminal angles have the same initial and terminal sides, so their difference is a multiple of \(360^\circ\). Since \(210^\circ-360^\circ=-150^\circ\), option A is correct. The difference between \(210^\circ\) and \(150^\circ\) is only \(60^\circ\). Exam tip: add or subtract \(360^\circ\) to test coterminal angles.

(405^\circ=\frac{405\pi}{180}=\frac{9\pi}{4}). Use the same rule even for angles greater than (360^\circ).

( \frac{3\pi}{8}\times \frac{180^\circ}{\pi}=67.5^\circ). Divide (180^\circ) by (8) and then multiply.

( \frac{5\pi}{24}\times \frac{180^\circ}{\pi}=37.5^\circ). If the denominator is (24), calculate (180^\circ\div 24).

( -\frac{7\pi}{6}\times \frac{180^\circ}{\pi}=-210^\circ). The negative sign in radians remains in degrees.

( \frac{13\pi}{8}\times \frac{180^\circ}{\pi}=292.5^\circ). Remember ( \frac{\pi}{8}=22.5^\circ ).