Angles

Subtract two full turns: \(765^\circ-2(360^\circ)=45^\circ\). Since \(45^\circ\) lies between \(0^\circ\) and \(90^\circ\), the terminal side is in the first quadrant.

For an angle in radians, arc length satisfies \(s=r\theta\). Thus \(\theta=\frac{s}{r}=\frac{11\text{ cm}}{14\text{ cm}}=\frac{11}{14}\) radians.

Use \(s=r\theta\) for an angle in radians: \(s=5\text{ cm}\times\frac{7\pi}{10}=\frac{7\pi}{2}\text{ cm}\).

Add 3 × 360° to get the coterminal angle −940° + 1080° = 140°. Since 90° < 140° < 180°, its terminal side lies in the second quadrant.

To find the principal angle, reduce the angle to the interval \([0,2\pi)\). Since \(2\pi=\frac{8\pi}{4}\), \(\frac{13\pi}{4}-2\pi=\frac{13\pi}{4}-\frac{8\pi}{4}=\frac{5\pi}{4}\). Hence, the principal angle is \(\frac{5\pi}{4}\). \(\frac{\pi}{4}\) is the reference angle, not the principal angle. Exam tip: add or subtract multiples of \(2\pi\) until the angle lies in \([0,2\pi)\).

For a sector, A = ½r²θ when θ is in radians. Thus 24 = ½ × 6² × θ = 18θ, giving θ = 24/18 = 4/3 radians.

Convert degrees to radians by multiplying by π/180: 225° × π/180° = 225π/180 = 5π/4 radians.

The direct answer is option B: 108°. The conversion rule is that a full turn is both 360° and 2π radians, so 1 radian equals 180° divided by π. Therefore, for the given angle, \\(\frac{3\pi}{5}\times\frac{180^\circ}{\pi}=\frac{3\times180^\circ}{5}=108^\circ\\). The π cancels first, then multiplication and division are easy. Option A, 96°, is wrong because it does not result from the conversion calculation. Option B, 108°, is correct because it follows directly from the standard radian-to-degree factor. Option C, 120°, would correspond to a different radian value, namely about \(\frac{2\pi}{3}\), not \(\frac{3\pi}{5}\). Option D, 135°, is also too large and would correspond to \(\frac{3\pi}{4}\). The safest exam method is to multiply radians by \(\frac{180^\circ}{\pi}\), cancel π, and simplify. Never multiply by 180° without dividing by π.

One complete rotation is 2π radians. Therefore, 8π ÷ 2π = 4 complete rotations.

The minute hand completes 2π radians in 60 minutes. In 17 minutes, it rotates (17/60) × 2π = 17π/30 radians.

Since 60 minutes make 1 degree, 15′ = 15/60° = 0.25°. Therefore, 11° 15′ = 11.25°.

Since π/2 ≈ 1.57 and π ≈ 3.14, we have π/2 < 2.35 < π. An angle in this range lies in the second quadrant.

The direct answer is B, \\(215^\circ\\). A reference angle is the small positive angle between the terminal arm and the nearest x-axis. In the third quadrant, an angle lies between \\(180^\circ\\) and \\(270^\circ\\), and the reference angle is measured after \\(180^\circ\\). Therefore, \\(\theta=180^\circ+35^\circ=215^\circ\\). Option A, \\(145^\circ\\), is \\(180^\circ-35^\circ\\), which belongs to the second quadrant. Option B is correct because 215° is in the third quadrant and has reference angle 35°. Option C, \\(325^\circ\\), is \\(360^\circ-35^\circ\\), so it belongs to the fourth quadrant. Option D, \\(305^\circ\\), also lies in the fourth quadrant and does not have the required reference angle. Exam cue: for a third-quadrant angle use \\(180^\circ+\alpha\\), not subtraction.

In the fourth quadrant, the principal angle is 360° minus the reference angle. Thus θ = 360° − 22° = 338°.

Subtract one full rotation, 2π = 12π/6: 19π/6 − 12π/6 = 7π/6. Since π < 7π/6 < 3π/2, the terminal side lies in the third quadrant.

One arcminute is 1/60 of a degree. Therefore, 1′ = (1/60)(π/180) = π/10800 radians.

One arcsecond is 1/60 of an arcminute, and one arcminute is 1/60 of a degree. Thus 1″ = (1/3600)° = π/(180 × 3600) = π/648000 radians.

Subtract 5 full turns, or 10π = 30π/3, from 31π/3. The result is π/3, which lies in [0, 2π).

Convert the angle to radians: 80° = 4π/9. The arc length is s = rθ = 9 × 4π/9 = 4π cm.

Convert the angle to radians: 150° = 5π/6. The sector area is A = ½r²θ = ½ × 12² × 5π/6 = 60π cm².

One complete revolution is 2π radians. Therefore, 45 revolutions correspond to 45 × 2π = 90π radians.

Since π ≈ 3.14 and 3π/2 ≈ 4.71, we have π < 3.9 < 3π/2. Angles in this interval lie in the third quadrant.

Subtract 2π to get the coterminal angle 13π/8, which lies in the fourth quadrant. Its reference angle is 2π − 13π/8 = 3π/8.

Since 30′ = 30/60° = 0.5°, the angle is 132.5° = 265/2°. Converting to radians gives (265/2)(π/180) = 53π/72.

The direct answer is A, \\(80^\circ\\). Coterminal angles differ by a whole turn, \\(360^\circ\\). The given principal angle is \\(280^\circ\\), so the angle represented by \\(-\theta\\) is \\(-280^\circ\\). To express it between \\(0^\circ\\) and \\(360^\circ\\), add one full turn: \\(-280^\circ+360^\circ=80^\circ\\). Option A is correct. Option B, \\(100^\circ\\), would result from using the wrong subtraction, not from negating 280°. Option C, \\(260^\circ\\), is not coterminal with \\(-280^\circ\\), because their difference is not a multiple of 360°. Option D, \\(300^\circ\\), is also unrelated to the required rotation. Geometrically, changing an angle’s sign reflects its terminal arm across the x-axis; the terminal arm at 280° reflects to 80°. Memory cue: for \\(-\theta\\), use \\(360^\circ-\theta\\) when \\(0^\circ<\theta<360^\circ\\).