Angles

To find a coterminal angle, subtract a multiple of \(360^\circ\) from the given angle. \(985^\circ-2\times360^\circ=985^\circ-720^\circ=265^\circ\). Therefore, \(265^\circ\) is the coterminal angle of \(985^\circ\) between \(0^\circ\) and \(360^\circ\). The other options do not differ from \(985^\circ\) by a whole multiple of \(360^\circ\). Exam tip: reduce an angle modulo \(360^\circ\) to obtain its standard coterminal angle.

(1540^\circ-1440^\circ=100^\circ). The principal angle is usually taken between (0^\circ) and (360^\circ).

The direct answer is option A, \(\frac{5\pi}{4}\). Coterminal angles differ by complete turns of \(2\pi\). We subtract enough complete turns to place the angle between 0 and \(2\pi\): \(\frac{29\pi}{4}-2\pi=\frac{21\pi}{4}\), and subtract \(2\pi\) again to get \(\frac{13\pi}{4}\), and once more to get \(\frac{5\pi}{4}\). Equivalently, \(2\pi=\frac{8\pi}{4}\), so \(\frac{29\pi}{4}-3\cdot\frac{8\pi}{4}=\frac{5\pi}{4}\), which lies in the required interval. Option A is correct. Option B, \(\frac{3\pi}{4}\), would result from subtracting too much or using an incorrect number of quarter-turns. Option C, \(\frac{7\pi}{4}\), is another possible-looking angle but is not coterminal with the given angle. Option D, \(\frac{\pi}{4}\), is also not obtained by subtracting a whole number of full turns. Remember: reduce by multiples of \(2\pi\), not by arbitrary fractions of a turn.

( -\frac{19\pi}{6}+\frac{24\pi}{6}=\frac{5\pi}{6} ). Add multiples of (2\pi) to a negative radian angle.

The governing concept is reduction of a radian measure to its principal coterminal angle. Coterminal angles differ by an integral multiple of 2π, since 2π radians represents one complete revolution. The principal representative is normally chosen in the interval 0 ≤ θ < 2π. Rewrite one revolution as 2π = 6π/3. Since 31π/3 = 30π/3 + π/3 = 5(2π) + π/3, removing five complete revolutions leaves π/3. This value is positive and smaller than 2π, so it satisfies the principal-angle interval. Therefore option B is correct. The other choices are different positions within the same interval and are not obtained from 31π/3 by subtracting an integral multiple of 2π; hence they are not coterminal with the given angle.

( -\frac{25\pi}{8}+\frac{32\pi}{8}=\frac{7\pi}{8} ). Keeping the same denominator is a safe method while adding (2\pi).

( -250^\circ+360^\circ=110^\circ ) and (110^\circ) lies in the second quadrant. First find the positive coterminal angle.

( -710^\circ+720^\circ=10^\circ ), so the terminal side lies in the first quadrant. Add multiples of (360^\circ) to a large negative angle.

( \pi<\frac{9\pi}{7}<\frac{3\pi}{2} ), so it lies in the third quadrant. Identify the quadrant directly from radian intervals.

( \pi<\frac{13\pi}{9}<\frac{3\pi}{2} ), so it is in the third quadrant. The interval between ( \pi ) and ( \frac{3\pi}{2} ) is the third quadrant.

( \frac{3\pi}{2}<\frac{17\pi}{9}<2\pi ), so it lies in the fourth quadrant. Compare radians by using a common denominator.

(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.

Convert minutes to fractional degrees using \(1^\circ=60'\).
\[76^\circ\ 45' = 76^\circ + \dfrac{45'}{60'}\, ^\circ = 76^\circ + 0.75^\circ = 76.75^\circ\]
So the decimal degree value is \(76.75^\circ\).

(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.

To convert degrees into radians, use the fact that \(180^\circ=\pi\) radians. Therefore an angle measured in degrees is multiplied by \(\frac{\pi}{180}\). For this angle, \(150^\circ\times\frac{\pi}{180}=\frac{150\pi}{180}=\frac{5\pi}{6}\), after cancelling the common factor 30. Hence the correct choice is B.

The proposed expression \(\frac{2\pi}{3}\) is not equal to 150 degrees. Converting it back gives \(\frac{2\pi}{3}\times\frac{180^\circ}{\pi}=120^\circ\). This identifies why it is incorrect. The other choices represent different angles: \(\frac{3\pi}{4}\) is 135 degrees, while \(\frac{7\pi}{6}\) and \(\frac{4\pi}{3}\) are larger than 180 degrees.

( \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.

The governing formula for the length of an arc is s = rθ when the central angle θ is measured in radians. The given angle is already in radians, so no degree-to-radian conversion is needed. Substitute r = 15 cm and θ = 2π/5: s = 15 × 2π/5 cm. Simplifying 15/5 to 3 gives s = 3 × 2π = 6π cm. Therefore option B is correct. The unit remains centimetres because the radius is measured in centimetres. The distractors 4π, 8π, and 10π cm result from using an incorrect factor, mishandling the denominator, or failing to simplify the product correctly. The formula is valid because the radian measure represents the ratio of arc length to radius.