Angles

Correct answer: C. (15) cm

Sector area is ( \frac{1}{2}rs ), so (150=\frac{1}{2}\times r\times20) gives (r=15). When arc length is given, ( \frac{1}{2}rs ) is useful.

Correct answer: B. ( \frac{2\pi}{5} )

( \frac{3\pi}{5} ) lies in the second quadrant, and the reference angle is ( \pi-\frac{3\pi}{5}=\frac{2\pi}{5} ). In the second quadrant use ( \pi-\theta ).

Correct answer: A. ( \frac{\pi}{7} )

( \frac{8\pi}{7} ) lies in the third quadrant, and ( \frac{8\pi}{7}-\pi=\frac{\pi}{7} ). In the third quadrant use ( \theta-\pi ).

Correct answer: A. ( \frac{\pi}{6} )

( \frac{13\pi}{6}-2\pi=\frac{\pi}{6} ), and it is in the first quadrant. First find the principal angle and then the reference angle.

Correct answer: C. (60^\circ)

(300^\circ) lies in the fourth quadrant, and (360^\circ-300^\circ=60^\circ). In the fourth quadrant use (360^\circ-\theta).

Correct answer: C. (60^\circ)

(240^\circ) lies in the third quadrant, and (240^\circ-180^\circ=60^\circ). In the third quadrant subtract (180^\circ).

Correct answer: B. (720^\circ)

(720^\circ=2\times360^\circ), so its terminal side lies on the positive (x)-axis. Multiples of (360^\circ) return to this axis.

Correct answer: B. (630^\circ)

(630^\circ-360^\circ=270^\circ), and (270^\circ) lies on the negative (y)-axis. First find the coterminal angle.

Correct answer: D. Negative (y)-axis

( \frac{15\pi}{2}-6\pi=\frac{3\pi}{2} ), and ( \frac{3\pi}{2} ) lies on the negative (y)-axis. Subtract multiples of (2\pi) to identify the axis.

Correct answer: C. They are coterminal

(1080^\circ=3\times360^\circ), so such angles are coterminal. If the difference is a multiple of (360^\circ), the terminal side is the same.

Correct answer: B. π

The governing idea is the standard position of an angle. An angle is measured anticlockwise from the positive x-axis. A complete revolution measures 2π radians, and the negative x-axis is reached after half a revolution. Therefore, its principal angle in the interval 0 < θ < 2π is π radians. The terminal sides of π and 2π are different: π ends on the negative x-axis, whereas 2π returns to the positive x-axis. Likewise, π/2 ends on the positive y-axis and 3π/2 ends on the negative y-axis. Hence option B is correct. The axes are boundary lines, not quadrants, so this is an axis-location question rather than a quadrant-sign question.

Correct answer: D. 70°

The governing concept is the reference angle, defined as the positive acute angle between an angle’s terminal side and the x-axis. First find a coterminal angle in the standard interval from 0° to 360°: −1190° + 4(360°) = 250°. The angle 250° lies in quadrant III. In quadrant III, the reference angle is measured from 180°, so it is 250° − 180° = 70°. Therefore option D is correct. The values 10°, 20°, and 30° are not obtained from the coterminal-angle calculation. Adding 360° only once or three times would leave a negative angle, so adding four full revolutions is appropriate. The corrected option set now contains one unambiguous answer.

Correct answer: A. 51°15′

The governing fact is that complementary angles have a sum of 90°. Therefore the required angle is 90° − 38°45′. Because the minutes cannot be subtracted directly from zero, rewrite 90° as 89°60′. Now subtract degrees and minutes separately: 89°60′ − 38°45′ = 51°15′. Hence option A is correct. The result can be checked by addition: 38°45′ + 51°15′ = 89°60′ = 90°. Option B reflects an incorrect borrowing step, option C mishandles the degree-minute conversion, and option D has the wrong number of minutes. Correct treatment of 60 minutes per degree makes the answer unique.

Correct answer: C. 29π/18

Direct answer: C, 29π/18. First substitute θ = 25π/18: π − θ = 18π/18 − 25π/18 = −7π/18. This value is negative, so it is not in the required interval from 0 to 2π. Add one complete revolution, 2π = 36π/18: −7π/18 + 36π/18 = 29π/18. This lies between 0 and 36π/18, so it is the required principal angle. B, 7π/18, is only the positive magnitude of the negative intermediate value; changing the sign changes the angle. A, 11π/18, and D, 31π/18, are not obtained by a valid integer multiple of 2π. Memory cue: when a reduced angle is negative, add 2π, not its absolute value.

Correct answer: D. 7π/4

The governing concept is that coterminal angles differ by an integer multiple of one complete revolution, 2π. We must add a suitable multiple of 2π to −17π/4 until the result lies in the standard interval 0 ≤ θ < 2π. Since 2π = 8π/4, adding 2π once gives −17π/4 + 8π/4 = −9π/4, which is still negative. Adding 2π a second time gives −π/4, still negative. Adding it a third time gives −17π/4 + 24π/4 = 7π/4. This value is positive and less than 2π, so it is the required representative. Therefore option D is correct. The other choices do not result from adding an integer multiple of 2π to the given angle.

Correct answer: B. Second quadrant

The governing idea is reduction by complete revolutions. A full revolution is 2π, which equals 10π/5. Subtracting two complete revolutions, 4π = 20π/5, from the given angle gives 23π/5 − 20π/5 = 3π/5. We now compare this reduced angle with the quadrant boundaries: π/2 = 2.5π/5 and π = 5π/5. Since 2.5π/5 < 3π/5 < 5π/5, the terminal arm lies between π/2 and π, which is the second quadrant. Thus option B is correct. The angle is not in the third quadrant because it is less than π, and it is not in the first or fourth quadrant because it is greater than π/2 and less than π. The extra full revolutions do not change the terminal arm.

Correct answer: B. 81 cm²

For a sector with its central angle measured in radians, the governing relations are s = rθ for arc length and A = 1/2 r²θ for area. The given values are s = 18 cm and θ = 2, so the radius is r = s/θ = 18/2 = 9 cm. Substituting into the area formula gives A = 1/2 × 9² × 2 = 1/2 × 81 × 2 = 81 cm². A useful independent check is A = 1/2 rs, which gives 1/2 × 9 × 18 = 81 cm² as well. Therefore option B is correct. The other options can arise from using the wrong radius, omitting the factor 1/2, or making an arithmetic error. The use of radians is essential for these compact formulas.

Correct answer: A. 8π/9

A reference angle is the positive acute angle between a terminal arm and the nearest x-axis. In the second quadrant, the standard angle is measured from the positive x-axis up to a point before π, so the angle equals π minus its reference angle. Let α = π/9. Then θ = π − α = π − π/9 = 8π/9. This value lies between π/2 and π, confirming that it is in the second quadrant. It also lies in the usual principal interval 0 ≤ θ < 2π, so no further reduction is required. Hence option A is correct. The value 10π/9 lies in the third quadrant, 17π/9 lies in the fourth quadrant, and 7π/9 does not equal π − π/9. The quadrant information is what determines whether the reference angle is added to or subtracted from π.

Correct answer: C. 10π square cm

For a sector whose central angle is expressed in radians, the governing area formula is A = 1/2 r²θ. Here r = 6 cm and θ = 5π/9 radians. Substitution gives A = 1/2 × 6² × 5π/9 = 1/2 × 36 × 5π/9 = 18 × 5π/9 = 10π square centimetres. Therefore option C is correct. It is important not to use θ/360 unless the angle is in degrees; the radian formula already incorporates the correct fraction of a full circle. Option A, B, or D could result from mishandling the factor 1/2, the radius squared, or the fraction 5/9. The unit must be square centimetres because the quantity calculated is an area.

Correct answer: B. 8/7 radian

The governing concept is the relation between a sector’s perimeter, its arc length, and radian measure. If r is the radius, θ is the central angle in radians, and s is the arc length, then s = rθ. The perimeter consists of two radii and one arc, so P = 2r + s = 2r + rθ. Substituting P = 22 and r = 7 gives 22 = 14 + 7θ. Hence 7θ = 8 and θ = 8/7 radians. Therefore option B is correct. The answer is not obtained by dividing the entire perimeter by the radius, because the two straight radii are not part of the arc. The other fractions fail when substituted into 14 + 7θ = 22.

Correct answer: D. They are coterminal

Two angles are coterminal when their difference is an integral multiple of 2π. Calculate the difference: 13π/5 − (−7π/5) = 20π/5 = 4π = 2(2π). Since 4π is a whole-number multiple of one complete revolution, both angles have the same terminal side. Hence option D is correct.

Correct answer: B. 6

The governing concept is the sector-area relation written using arc length. For a sector with radius r and arc length s, its area is A = 1/2 rs. The statement says A = 3s, so equate the two expressions: 1/2 rs = 3s. Since a genuine sector has positive arc length, divide both sides by s to obtain r/2 = 3, and therefore r = 6. Option B is correct. The central angle is unnecessary because it is already included in the arc length relation s = rθ. A common mistake is to omit the factor 1/2, which changes the result. The other numerical options do not satisfy A = 3s when the correct sector formula is used.

Correct answer: A. 1°

The governing concept is the continuous angular motion of both clock hands. The minute hand moves 6° per minute, so at 38 minutes its position is 38 × 6 = 228°. The hour hand moves 30° per hour and also advances 0.5° per minute. At 7:38, its position is 7 × 30 + 38 × 0.5 = 210 + 19 = 229°. The difference is |229 − 228| = 1°. Since this difference is already less than 180°, it is the smaller angle. Therefore option A is correct. Treating the hour hand as fixed exactly at 7 would ignore its 19° movement and produce a misleading result. The calculation must account for both hands’ positions at the same instant.

Correct answer: C. 9π/2

The governing concept is uniform angular motion: one complete revolution equals 2π radians. A second hand completes one revolution in 60 seconds, so its angular speed is 2π/60 = π/30 radians per second. The elapsed time is 2 minutes 15 seconds, which is 120 + 15 = 135 seconds. Hence the angle turned is θ = (π/30) × 135 = 135π/30 = 9π/2 radians. Therefore option C is correct. The same result follows by noting that 135 seconds represents 135/60 = 9/4 revolutions, and (9/4) × 2π = 9π/2. Options A, B, and D arise from incorrect time conversion or an incorrect revolution count.

Correct answer: B. 2π/5

The governing concept is the reference angle, defined as the positive acute angle between a terminal side and the x-axis. To locate 7π/5, compare it with π = 5π/5 and 3π/2 = 7.5π/5. Thus 7π/5 lies in the third quadrant. For a third-quadrant angle θ, the reference angle is θ − π. Therefore α = 7π/5 − 5π/5 = 2π/5. This value is positive and less than π/2, so it satisfies the definition. Hence option B is correct. The expression 2π − θ belongs to the fourth-quadrant case and would be inappropriate here; the other options do not give the required acute angle.