Angles

Correct answer: C. 105

Convert radians to degrees by multiplying by 180/π: (7π/12) × (180/π) = 105°. Therefore, a = 105.

Correct answer: A. 16 + 6π cm

The arc length is s = rθ = 8 × 3π/4 = 6π cm. A sector’s perimeter includes the arc and two radii, so it is 6π + 2(8) = 16 + 6π cm.

Correct answer: A. They are coterminal

Their difference is 410° − (−310°) = 720° = 2 × 360°. Since coterminal angles differ by an integer multiple of 360°, the angles are coterminal.

Correct answer: A. 4π/9

The hour hand completes 2π radians in 12 hours, so its rate is π/6 radians per hour. The elapsed time is 2 + 40/60 = 8/3 hours. Thus, it rotates through (8/3)(π/6) = 4π/9 radians.

Correct answer: A. π/6

Subtract 6π, or 36π/6, to find a coterminal angle: 37π/6 − 36π/6 = π/6. This lies in the first quadrant, so the reference angle is π/6.

Correct answer: A. 15/4

One revolution is 2π radians. Therefore, x = (15π/2) ÷ 2π = 15/4 revolutions.

Correct answer: C. 15 cm

For an angle in radians, the arc length is s = rθ. Hence r = s/θ = 10π ÷ (2π/3) = 10π × 3/(2π) = 15 cm.

Correct answer: B. (2)

The direct answer is option B: k = 2. To convert degrees to radians, multiply by \(\frac{\pi}{180^\circ}\). Thus, \(72^\circ\times\frac{\pi}{180^\circ}=\frac{72\pi}{180}=\frac{2\pi}{5}\), after dividing numerator and denominator by 36. Comparing this result with the required form \(\frac{k\pi}{5}\), the numerator shows that k = 2. Option A, 1, is wrong because \(\frac{\pi}{5}\) equals 36°, not 72°. Option B, 2, is correct because \(\frac{2\pi}{5}\) equals 72°. Option C, 3, would give \(\frac{3\pi}{5}=108^\circ\), which is too large. Option D, 4, would give \(\frac{4\pi}{5}=144^\circ\), also too large. Always simplify the fraction after converting; the coefficient of π is the required k. A useful memory pair is 180° = π radians, so degrees-to-radians uses π/180.

Correct answer: C. Third quadrant

Convert to degrees: (11π/9)(180°/π) = 220°. Since 180° < 220° < 270°, the angle lies in the third quadrant.

Correct answer: B. 60°

Subtract 1440° (four full revolutions) to get the coterminal angle 120°. It lies in the second quadrant, so its reference angle is 180° − 120° = 60°.

Correct answer: D. Fourth quadrant

Add 2π = 12π/6 twice to obtain a coterminal angle: −25π/6 + 24π/6 = −π/6. Adding another 2π gives 11π/6, which lies in the fourth quadrant.

Correct answer: A. Coterminal

Their difference is 19π/7 − 5π/7 = 14π/7 = 2π, one full revolution. Angles that differ by an integer multiple of 2π are coterminal.

Correct answer: C. 49

Convert degrees to radians: \(245°\times\frac{\pi}{180°}=\frac{49\pi}{36}\). Comparing this with \(\frac{m\pi}{36}\) gives \(m=49\).

Correct answer: B. \(\frac{5\pi}{12}\)

Since \(\pi<\frac{17\pi}{12}<\frac{3\pi}{2}\), the angle lies in quadrant III. Its reference angle is the difference from \(\pi\): \(\frac{17\pi}{12}-\pi=\frac{5\pi}{12}\).

Correct answer: B. 76°

Convert radians to degrees: \(\frac{4}{3}\times\frac{180°}{\pi}=\frac{240°}{\pi}\approx76.4°\). Rounded to the nearest whole degree, this is 76°.

Correct answer: A. 279°, fourth quadrant

Subtract two full rotations: \(999°-2(360°)=279°\). Since \(270°<279°<360°\), the principal angle lies in the fourth quadrant.

Correct answer: B. \(\frac{14\pi}{3}\) cm

First convert the angle to radians: \(40°=\frac{2\pi}{9}\). The arc length is \(s=r\theta\), so \(s=21\times\frac{2\pi}{9}=\frac{14\pi}{3}\) cm.

Correct answer: C. (\frac{5\pi}{6})

Multiply degrees by ( \frac{\pi}{180} ) to convert into radians. In exams identify multiples of (30^\circ) quickly.

Correct answer: B. (150^\circ)

(-\frac{7\pi}{6}=-210^\circ) and adding (360^\circ) gives (150^\circ). For a negative angle add (360^\circ) to get the principal angle.

Correct answer: C. \(200^\circ\)

Coterminal angles are obtained by adding or subtracting integral multiples of \(360^\circ\). Here, \(920^\circ-2\times360^\circ=920^\circ-720^\circ=200^\circ\). Since it is positive and less than \(360^\circ\), \(200^\circ\) is the least positive coterminal angle. \(220^\circ\) would require subtracting \(700^\circ\), which is not a multiple of \(360^\circ\). Exam tip: reduce a degree measure modulo \(360^\circ\).

Correct answer: B. \(33^\circ\)

Let the angle be \(x^\circ\). Its complement is \((90-x)^\circ\). Given that the complement is \(24^\circ\) greater than the angle, \(90-x=x+24\). Thus, \(2x=66\), so \(x=33\). Therefore, the correct answer is \(33^\circ\). For example, if the angle were \(36^\circ\), its complement would be \(54^\circ\), which is only \(18^\circ\) greater. Exam tip: complementary angles always add up to \(90^\circ\).

Correct answer: C. (45^\circ)

If the angle is (x) then (180^\circ-x=3x) gives (x=45^\circ). Supplementary angles add to (180^\circ).

Correct answer: B. \(\frac{11\pi}{2}\)

One complete revolution equals \(2\pi\) radians. Since \(2.75=\frac{11}{4}\), the angle is \(\frac{11}{4}\times 2\pi=\frac{11\pi}{2}\) radians. Note that \(\frac{9\pi}{2}\) represents only \(2.25\) revolutions, so it is not correct. Exam tip: multiply the number of revolutions by \(2\pi\) to convert revolutions into radians.

Correct answer: C. \(6\pi\) cm

Since the central angle is given in radians, use the arc-length formula \(s=r\theta\). Here, \(r=14\) cm and \(\theta=\frac{3\pi}{7}\). Thus, \(s=14\times\frac{3\pi}{7}=6\pi\) cm. \(7\pi\) cm would result from incorrect simplification of the product. Exam tip: In \(s=r\theta\), the angle \(\theta\) must be in radians.

Correct answer: A. \(20+4\pi\) cm

The perimeter of a sector is the sum of its two radii and its arc length: \(P=2r+r\theta\), where \(\theta\) is in radians. Here, the arc length is \(10\times\frac{2\pi}{5}=4\pi\) cm. Therefore, \(P=2(10)+4\pi=20+4\pi\) cm. Option C incorrectly takes the arc length as \(2\pi\). Exam tip: when the angle is in radians, use \(r\theta\) directly for arc length.