Angles

The governing conversion relationship is 180° = π radians. Hence a degree measure is converted to radians by multiplying by π/180. For 345°, the calculation is 345 × π/180 = (345/180)π. Dividing numerator and denominator by 15 gives 23π/12 radians, so option C is correct. A magnitude check also supports this result: 345° is only 15° less than 360°, so its radian measure should be slightly less than 2π; indeed, 23π/12 = 2π − π/12. Option A equals 330°, option B equals 285°, and option D equals 315°, so none of those represents 345°.

Use the standard equivalence π radians = 180°. To convert 11π/12 radians to degrees, multiply by 180°/π: (11π/12) × (180°/π). The π factors cancel, leaving 11 × 180°/12 = 11 × 15° = 165°. Therefore option C is correct. The size is also sensible because 11π/12 is less than π but close to π, so its degree measure must be less than 180° but close to it. Option A is 150° and equals 5π/6. Option B is not the exact result of the conversion, while option D is 10° greater than the calculated value. Exact cancellation avoids approximation errors.

The governing concept is coterminal angles. Two angles are coterminal when their terminal sides coincide, so their difference must be an integral multiple of one full revolution, 360°. A principal angle in degrees is usually chosen in the interval 0° ≤ θ < 360°. Reduce 1125° by subtracting complete revolutions: 1125° − 3 × 360° = 1125° − 1080° = 45°. Thus 1125° and 45° have the same terminal side, making option C correct. The alternatives 15°, 30°, and 75° do not differ from 1125° by an integral multiple of 360°, so they represent different terminal positions. The calculation also confirms that the result lies in the required principal interval.

The governing concept is coterminal angles measured in radians. One complete revolution equals 2π radians, so coterminal angles differ by an integral multiple of 2π. The principal representative is normally selected in 0 ≤ θ < 2π. Since 14π/3 is larger than 2π, subtract two complete revolutions: 14π/3 − 4π = 14π/3 − 12π/3 = 2π/3. This value lies in the principal interval, so option C is correct. Equivalently, subtracting 2π once gives 8π/3, which is still too large, and subtracting it twice gives 2π/3. The other listed values have different terminal sides and are not equivalent to the given angle.

Answer: option A, 143.2°. The basic conversion is π radians = 180°, so 1 radian = 180°/π. To convert 2.5 radians, multiply by 180°/π: 2.5 × 180°/π = 450°/π. Using π ≈ 3.1416 gives 450/3.1416 ≈ 143.24°, which is nearest to 143.2°. A quick check uses 1 radian ≈ 57.3°, so 2.5 × 57.3° ≈ 143.25°. Option B, 125.0°, is about 18.2° away; option C, 157.5°, is about 14.3° away; and option D, 172.0°, is about 28.8° away. Thus A is the closest value. The common mistake is reversing the factor: π/180 is for degrees to radians, whereas 180/π is for radians to degrees. Memory cue: radians-to-degrees means multiply by about 57.3.

The governing idea is the division of one full revolution into four quadrants using the standard boundary angles 0°, 90°, 180°, 270°, and 360°. Compare 135° with the consecutive boundaries around it: 90° < 135° < 180°. Therefore θ lies between 90° and 180° and is located in the second quadrant, so option B is correct. It cannot be in the first quadrant because that interval ends at 90°. It cannot be in the third quadrant because that begins at 180°, and it cannot be in the fourth quadrant because that begins at 270°. The simple inequality is sufficient; no trigonometric calculation is required.

For a circular arc measured with the central angle in radians, the governing formula is s = rθ, where s is arc length, r is radius, and θ is the angle in radians. Rearranging gives θ = s/r. Substituting the given measurements, θ = 14 cm / 7 cm = 2. The centimetre units cancel, leaving a dimensionless radian measure. Therefore option B, 2 radians, is correct. Choosing 1 would incorrectly divide by 14, while 7 and 14 confuse the radius or arc length with the required angle. The formula applies directly because the angle is requested in radians.

The governing concept is that coterminal angles differ by an integral multiple of 360°. We need to add enough complete turns to the negative angle so that the result lies in the interval from 0° to 360°. Add 360° three times: -1000° + 360° = -640°, then -640° + 360° = -280°, and finally -280° + 360° = 80°. Equivalently, -1000° + 1080° = 80°, where 1080° is 3 × 360°. Since 80° lies in the required interval, option B is correct. Options A, C, and D do not result from the required 360° adjustment and therefore are not coterminal with -1000°.

A principal coterminal angle in this question is the representative between 0° and 360°. Coterminal angles are obtained by adding or subtracting integer multiples of 360°. Starting with -1430°, add 4 × 360° = 1440°: -1430° + 1440° = 10°. The result is positive and is less than 360°, so it satisfies the required principal interval. Therefore, option D is correct. The tempting values 350°, 20°, and 30° are not equal to -1430° plus an integral multiple of 360°. For example, the nearest adjustment using 3 turns gives -350°, not a principal positive representative; one additional turn gives 10°.

The governing conversion is 1 degree = 60 minutes. Separate the whole-degree part from the fractional part: 19.2° = 19° + 0.2°. Only the fractional degree must be converted into minutes. Multiply it by 60: 0.2 × 60′ = 12′. Therefore, 19.2° = 19° 12′, so option C is correct. Option A incorrectly changes 0.2 into 2 minutes, and option B treats the decimal as if it represented 20 minutes without performing the conversion. Option D incorrectly changes the whole-degree part from 19° to 20°. A reverse check confirms the result because 12′ = 12/60° = 0.2°, and 19° + 0.2° returns 19.2° exactly.

Answer: option C, 7π/10 radians. The fundamental relation is 180° = π radians. Therefore, to convert a degree measure into radians, multiply it by π/180. For 126°, we get 126 × π/180 = (126/180)π. Divide numerator and denominator by 18: 126/180 = 7/10. Hence 126° = 7π/10 radians. Option A, 5π/7, is not the simplified form of 126π/180. Option B, 3π/4, represents 135°, because (3π/4)×(180/π)=135°. Option C gives (7π/10)×(180/π)=126°, so it is correct. Option D also does not simplify from 126/180. The common confusion is reversing the conversion factor: π/180 changes degrees to radians, while 180/π changes radians to degrees. Memory cue: attach π/180 to a degree value.

Answer: option A, 9°. The basic conversion is π radians = 180°. Therefore, to convert radians into degrees, multiply by 180°/π. For π/20 radians, the calculation is (π/20) × (180°/π). The π in the numerator and denominator cancels, leaving 180°/20 = 9°. Thus π/20 radians equals 9°, so option A is correct. Option B, 10°, would come from dividing 180 by 18 rather than 20. Option C, 12°, corresponds to π/15 radians, not π/20. Option D, 18°, corresponds to π/10 radians. The safest method is to write the conversion factor with units so that radians cancel. Do not use π/180 here: that factor converts degrees to radians and would reverse the requested direction. Memory cue: a fraction of π radians is the same fraction of 180°; one-twentieth of 180° is 9°.

Direct answer: C, 58.2°. Degrees and minutes use a base-60 relationship, not a base-100 relationship. One degree equals 60 minutes, so the 12 minutes must first be converted into a fraction of a degree: 12′ = 12/60° = 0.2°. Add this fractional part to the whole-degree part: 58° + 0.2° = 58.2°. Therefore C is correct. A, 58.12°, incorrectly places the minute number after the decimal point as if 12 minutes meant 0.12 degree. B, 58.15°, would correspond to 9 minutes, not 12 minutes. D, 58.25°, would correspond to 15 minutes. Memory cue: for degrees and minutes, divide minutes by 60 before writing decimal notation.

To convert decimal degrees into degrees and minutes, separate the whole-number and fractional parts. The whole-number part of 84.4° is 84°, while the fractional part is 0.4°. Because 1° = 60′, multiply the fractional part by 60: 0.4 × 60′ = 24′. Therefore, 84.4° = 84° 24′, making option B correct. Option A results from writing the decimal digits as minutes without conversion. Option C incorrectly treats 0.4 as 40 minutes, and option D adds an unnecessary degree. Since the converted minute value is 24′, which is less than 60′, no carrying into the degree part is needed.

Coterminal angles have the same terminal side and differ by an integral multiple of 360°, since 360° represents one complete revolution. The question asks for the representative satisfying 0° ≤ θ < 360°. Subtract one complete revolution from 540°: 540° − 360° = 180°. This result is nonnegative and smaller than 360°, so it meets the required interval and is the desired value of θ. Therefore option C is correct. Subtracting 360° once more would produce −180°, which is outside the interval. The alternatives 90°, 120°, and 270° are not coterminal with 540°, because their differences from 540° are not integer multiples of 360°. Thus the principal degree representative is 180°.

The governing conversion facts are 60 angular minutes = 1 degree and 180° = π radians. First convert 3150′ into degrees: 3150 ÷ 60 = 52.5° = 105/2°. Next convert degrees to radians by multiplying by π/180. Thus the radian measure is (105/2) × (π/180) = 105π/360. Dividing numerator and denominator by 15 gives 7π/24. Therefore option B is correct. A useful check is that 3150′ is 52.5°, which is slightly less than 60°, so its radian value should be slightly less than π/3; 7π/24 is indeed 0.2917π, while the other choices represent different degree measures. The prime symbol denotes angular minutes, not ordinary length units.

The governing relation for arc length is s = rθ when θ is measured in radians. Solving for the central angle gives θ = s/r. With s = 8π cm and r = 16 cm, θ = 8π/16 = π/2 radians. Therefore option B is correct. We can verify the result geometrically: π/2 is one quarter of a full revolution, and one quarter of a circle of radius 16 has arc length (1/4)(2πr) = (1/4)(32π) = 8π cm, exactly as given. If θ were π/4, the arc would be 4π cm; if it were π, the arc would be 16π cm; and 3π/2 would give 24π cm. Hence only π/2 satisfies the data.

The governing fact is that the hour hand completes one full turn of 360° in 12 hours. Its angular speed is therefore 360°/12 = 30° per hour, or 0.5° per minute. The elapsed time is 3 hours 20 minutes = 3 × 60 + 20 = 200 minutes. Multiplying by the minute rate gives 200 × 0.5° = 100°. Equivalently, 3 hours 20 minutes is 10/3 hours, and 30 × 10/3 = 100°. Therefore option B is correct. The 90° option ignores the extra 20 minutes and uses only three hours. The 110° and 120° choices overestimate the motion; 120° would correspond to four complete hours. The calculation assumes the hour hand moves continuously, as a real clock does.

A radian is defined geometrically by a circle’s arc. If an arc has length s equal to the radius r, then the angle subtended at the centre is one radian. This follows from the relation s = rθ: setting s = r gives θ = 1. Therefore option A is correct. An arc equal to the diameter does not define one radian; it would give θ = 2 radians. An angle of 90° is π/2 radians, not 1 radian, and an arc equal to the full circumference subtends 2π radians, a complete revolution. Thus the key condition is equality of arc length and radius, measured in the same units.

The governing concept is conversion between radians and degrees. The fundamental equivalence is π radians = 180°. Therefore, to convert 11π/36 radians into degrees, multiply by 180°/π: x = (11π/36) × (180°/π). The factor π cancels, leaving x = 11 × 180°/36 = 11 × 5° = 55°. Thus option D is correct. A useful check is that 11π/36 is slightly greater than π/4, and π/4 equals 45°, so a result somewhat greater than 45° is reasonable. The other options do not result from the correct conversion factor; confusing π radians with 90° or simplifying 180/36 incorrectly can lead to them.

The governing concept is the domain restriction of a reciprocal trigonometric function. Since cosec x = 1/sin x, it is undefined whenever the denominator sin x equals zero. The sine function is zero at 0, π, 2π, 3π and all corresponding negative multiples, which is written compactly as x = nπ, where n is any integer. Hence option B is correct. At x = π/2 + nπ, sin x is either 1 or −1, so cosec x is defined. The forms in options C and D do not represent all zeros of sine. The integer condition for n is essential because it describes the complete repeating set.