Degrees & Radians - Math 3

1. Two gears are connected such that when the smaller gear (radius 4 cm) rotates through $\frac{5\pi}{3}$ radians, the larger gear (radius 10 cm) rotates through angle $\theta$. What is $\theta$ in both radians and degrees?
   Answer: $\frac{2\pi}{3}$ radians, 120° When two gears are connected, they move together such that the distance traveled along their circumferences is equal. This means the arc length covered by each gear must be the same, which gives us the key relationship: $r_1 \theta_1 = r_2 \theta_2$. For the smaller gear, the arc length is $4 \times \frac{5\pi}{3} = \frac{20\pi}{3}$ cm. Since this same arc length must be covered by the larger gear, we have: $10 \times \theta = \frac{20\pi}{3}$. Solving for $\theta$: $\theta = \frac{20\pi}{3} \div 10 = \frac{20\pi}{30} = \frac{2\pi}{3}$ radians. To convert to degrees: $\frac{2\pi}{3} \times \frac{180°}{\pi} = \frac{2 \times 180°}{3} = 120°$. Answer A ($\frac{25\pi}{6}$ radians, 750°) likely comes from incorrectly multiplying instead of using the inverse relationship between gear sizes and rotation angles. Answer B ($\frac{4\pi}{3}$ radians, 240°) might result from using the wrong radius ratio or making an arithmetic error in the proportion. Answer D ($\frac{\pi}{2}$ radians, 90°) could come from confusion about the relationship or using an incorrect conversion factor. Remember that in connected gears, the smaller gear always rotates through a larger angle than the bigger gear for the same arc length. When you see gear problems, immediately think "equal arc lengths" and set up the equation $r_1 \theta_1 = r_2 \theta_2$ to find the unknown angle.
2. Two angles are complementary, and one angle measures $\frac{\pi}{9}$ radians. What is the measure of the other angle in degrees?
   Answer: 70° When you encounter complementary angles, remember that they always sum to 90° (or $\frac{\pi}{2}$ radians). This is a fundamental relationship that applies regardless of which units are used to measure the angles. Since one angle measures $\frac{\pi}{9}$ radians, you need to find its complement. The other angle equals $\frac{\pi}{2} - \frac{\pi}{9}$. To subtract these fractions, find a common denominator: $\frac{\pi}{2} = \frac{9\pi}{18}$ and $\frac{\pi}{9} = \frac{2\pi}{18}$. Therefore, the complement is $\frac{9\pi}{18} - \frac{2\pi}{18} = \frac{7\pi}{18}$ radians. Now convert to degrees using the relationship $180° = \pi$ radians. Multiply by $\frac{180°}{\pi}$: $\frac{7\pi}{18} \times \frac{180°}{\pi} = \frac{7 \times 180°}{18} = \frac{1260°}{18} = 70°$ This confirms answer choice D is correct. Choice A (75°) likely comes from incorrectly assuming complementary angles sum to 95° instead of 90°. Choice B (80°) might result from computational errors in the radian-to-degree conversion. Choice C (60°) could stem from confusing complementary angles (sum to 90°) with angles in an equilateral triangle (each 60°). Always remember the key relationships: complementary angles sum to 90°, and to convert radians to degrees, multiply by $\frac{180°}{\pi}$. When working with mixed units, convert everything to the same unit system before performing calculations.
3. A pendulum swings through an angle of $\frac{2\pi}{9}$ radians. If the pendulum arm is 18 inches long, what is the arc length of the swing in inches, and what is the swing angle in degrees?
   Answer: 4π inches, 40° This problem tests your understanding of arc length formulas and radian-to-degree conversions—two fundamental concepts that often appear together. To find the arc length, use the formula: arc length = radius × angle (in radians). With a radius of 18 inches and an angle of $\frac{2\pi}{9}$ radians, the calculation is: $18 \times \frac{2\pi}{9} = \frac{36\pi}{9} = 4\pi$ inches. For the degree conversion, use the relationship $\pi \text{ radians} = 180°$. Set up the proportion: $\frac{2\pi}{9} \text{ radians} \times \frac{180°}{\pi \text{ radians}} = \frac{2\pi \times 180°}{9\pi} = \frac{360°}{9} = 40°$. Looking at the wrong answers: Choice A correctly calculates the arc length as $4\pi$ inches but incorrectly converts to 45°—this happens when students mistakenly use $\frac{\pi}{4}$ radians instead of $\frac{2\pi}{9}$. Choice B makes the opposite error pattern, incorrectly calculating $6\pi$ inches (perhaps multiplying $18 \times \frac{\pi}{3}$) while also getting 45° for the angle. Choice D gets the angle conversion right at 40° but calculates the arc length as $3\pi$ inches, likely from incorrectly using $\frac{\pi}{6}$ radians or making an arithmetic error. The correct answer is C: $4\pi$ inches and 40°. Study tip: Always double-check both your radian-degree conversion and arc length calculation separately. These problems test two distinct skills, and it's easy to get one right while missing the other.
4. An angle measures $\frac{11\pi}{6}$ radians. What is the measure of its reference angle in degrees?
   Answer: $30°$ First, convert $\frac{11\pi}{6}$ to degrees: $\frac{11\pi}{6} \times \frac{180°}{\pi} = \frac{11 \times 180°}{6} = 330°$. Since $330°$ is in the fourth quadrant (between $270°$ and $360°$), the reference angle is $360° - 330° = 30°$. Choice B ($60°$) results from confusing this with $\frac{\pi}{3}$. Choice C ($150°$) incorrectly calculates as if the angle were in the second quadrant. Choice D ($330°$) gives the original angle in degrees, not the reference angle.
5. A wheel rotates through an angle of $\frac{7\pi}{4}$ radians. If the wheel then rotates an additional $315°$, what is the total angular displacement of the wheel in radians?
   Answer: $\frac{7\pi}{2}$ First, convert $315°$ to radians: $315° \times \frac{\pi}{180°} = \frac{315\pi}{180} = \frac{7\pi}{4}$ radians. The total displacement is $\frac{7\pi}{4} + \frac{7\pi}{4} = \frac{14\pi}{4} = \frac{7\pi}{2}$ radians. Choice B results from incorrectly converting $315°$ to $\frac{\pi}{4}$. Choice C comes from converting $315°$ to $\frac{15\pi}{4}$. Choice D results from adding $315$ as a pure number to $\frac{7\pi}{4}$.
6. On a unit circle, an arc of length $\frac{5\pi}{3}$ is measured from the positive x-axis. If this arc length is converted to degrees and then reduced to find the coterminal angle between $0°$ and $360°$, what is the result?
   Answer: $300°$ Convert $\frac{5\pi}{3}$ radians to degrees: $\frac{5\pi}{3} \times \frac{180°}{\pi} = \frac{5 \times 180°}{3} = 300°$. Since $300°$ is already between $0°$ and $360°$, no reduction is needed. Choice A ($60°$) results from calculating $\frac{\pi}{3}$ instead of $\frac{5\pi}{3}$. Choice B ($240°$) comes from incorrectly calculating $\frac{4\pi}{3}$. Choice D ($120°$) results from calculating $\frac{2\pi}{3}$.
7. A satellite orbits Earth in a circular path. During a 45-minute period, it travels through a central angle of $\frac{3\pi}{2}$ radians. If the orbital radius is 8000 km, what distance does the satellite travel during this time?
   Answer: 12,000π km When you encounter circular motion problems involving satellites or any object moving in a circular path, you need to connect the central angle (in radians) with the arc length using the fundamental relationship: arc length = radius × central angle. Here, the satellite travels through a central angle of $\frac{3\pi}{2}$ radians with an orbital radius of 8000 km. The distance traveled is simply the arc length along this circular path. Using the arc length formula: Distance = radius × central angle = 8000 km × $\frac{3\pi}{2}$ radians = $\frac{24,000\pi}{2}$ km = 12,000π km This confirms answer choice C is correct. Let's examine why the other answers are wrong. Answer A (16,000π km) would result from incorrectly using 2π as the central angle instead of $\frac{3\pi}{2}$, perhaps confusing this with a complete orbit. Answer B (6,000π km) comes from using $\frac{3\pi}{4}$ as the central angle, which is half the actual value given. Answer D (24,000π km) results from forgetting to divide by 2 in the final calculation—you'd get this if you calculated 8000 × 3π instead of 8000 × $\frac{3\pi}{2}$. Remember: for any circular motion problem, the key is identifying what information you have (radius and central angle here) and applying the correct formula. Always double-check that you're using radians correctly in your calculations, and be careful with fraction arithmetic in the final steps.
8. A circular garden path has radius 12 meters. A person walks along an arc that subtends a central angle of $\frac{7\pi}{6}$ radians. How far does the person walk, and what is this angle in degrees?
   Answer: 14π meters, 210° When you encounter arc length problems, you're working with the relationship between a circle's radius, central angle, and the distance along the curved path. The key formula is: arc length = radius × central angle (in radians). For the distance calculation, you multiply the radius (12 meters) by the central angle ($\frac{7\pi}{6}$ radians): $12 \times \frac{7\pi}{6} = \frac{84\pi}{6} = 14\pi$ meters. To convert radians to degrees, use the conversion factor $\frac{180°}{\pi}$: $\frac{7\pi}{6} \times \frac{180°}{\pi} = \frac{7 \times 180°}{6} = \frac{1260°}{6} = 210°$ Answer D (14π meters, 210°) is correct with both calculations. Answer A gives 10π meters, which would result from incorrectly using $\frac{5\pi}{6}$ as the angle instead of $\frac{7\pi}{6}$. Answer B shows 12π meters and 180°—the arc length uses the radius value instead of properly multiplying, and 180° corresponds to $\pi$ radians, not $\frac{7\pi}{6}$. Answer C has the correct arc length of 14π meters but incorrectly converts to 240°, which would be $\frac{4\pi}{3}$ radians. Remember: always keep your angle units consistent. Use radians for arc length calculations, then convert to degrees if needed. Double-check conversions by remembering that $\pi$ radians equals 180°, so angles greater than $\pi$ will exceed 180°.
9. A wheel with radius 5 inches rolls along a straight line. If the wheel makes $\frac{3}{4}$ of a revolution, what distance does the center of the wheel travel?
   Answer: $\frac{15\pi}{2}$ inches When a wheel rolls along a straight line, the distance the center travels equals the arc length that has "unrolled" along the ground. This is a fundamental relationship in rolling motion problems. To find how far the center travels, you need to calculate the arc length corresponding to the wheel's rotation. The circumference of this wheel is $2\pi r = 2\pi(5) = 10\pi$ inches. Since the wheel makes $\frac{3}{4}$ of a complete revolution, the arc length is $\frac{3}{4} \times 10\pi = \frac{30\pi}{4} = \frac{15\pi}{2}$ inches. This is the distance the center travels. Looking at the wrong answers: Choice A ($\frac{5\pi}{2}$) appears to come from incorrectly using just the radius times $\frac{3}{4}$ revolution times $\pi$, which misses the factor of 2 in the circumference formula. Choice B ($10\pi$) gives you the full circumference, suggesting the student forgot to multiply by the fraction $\frac{3}{4}$. Choice C ($\frac{15\pi}{4}$) is close but represents a calculation error—likely from computing $\frac{3}{4} \times 5\pi$ instead of $\frac{3}{4} \times 10\pi$. The correct answer is D: $\frac{15\pi}{2}$ inches. Study tip: For rolling wheel problems, always remember that distance traveled equals (fraction of revolution) × (circumference). The circumference formula is $2\pi r$, not just $\pi r$—this factor of 2 is the most common source of errors on these problems.
10. A circular sector has a central angle of $150°$ and a radius of 6 units. If the arc length of this sector equals the arc length of another sector with radius 4 units, what is the central angle of the second sector in radians?
    Answer: $\frac{5\pi}{4}$ When you encounter problems about circular sectors, remember that arc length depends on both the radius and the central angle. The arc length formula is $s = r\theta$, where $\theta$ must be in radians. Let's find the arc length of the first sector. Since the central angle is given as $150°$, we need to convert to radians: $150° \times \frac{\pi}{180°} = \frac{5\pi}{6}$ radians. With radius 6 units, the arc length is $s_1 = 6 \times \frac{5\pi}{6} = 5\pi$ units. Since both sectors have equal arc lengths, the second sector also has arc length $5\pi$ units. Using the arc length formula for the second sector: $5\pi = 4\theta_2$, where $\theta_2$ is the unknown central angle. Solving for $\theta_2$: $\theta_2 = \frac{5\pi}{4}$ radians. Looking at the wrong answers: Choice A ($\frac{3\pi}{2}$) would give an arc length of $4 \times \frac{3\pi}{2} = 6\pi$, which is too large. Choice B ($\frac{5\pi}{6}$) represents the first sector's angle in radians, not the second's. Choice C ($\frac{9\pi}{4}$) would produce an arc length of $4 \times \frac{9\pi}{4} = 9\pi$, far exceeding our target. The answer is D: $\frac{5\pi}{4}$. Study tip: Always convert degrees to radians before using the arc length formula, and remember that when radii differ but arc lengths are equal, the sector with the smaller radius must have the larger central angle.
11. On a unit circle, an arc subtends a central angle of $\frac{7\pi}{4}$ radians. What is the equivalent angle measure in degrees, and in which quadrant does the terminal side lie?
    Answer: 315°, Quadrant IV Convert: $\frac{7\pi}{4} \cdot \frac{180°}{\pi} = \frac{7 \cdot 180°}{4} = \frac{1260°}{4} = 315°$. Since $270° < 315° < 360°$, the terminal side lies in Quadrant IV. Choice B uses $\frac{19\pi}{12}$ radians instead. Choice C has correct degrees but wrong quadrant (confuses with $225°$). Choice D uses $\frac{23\pi}{12}$ radians.
12. A clock's minute hand has length 8 cm. Between 2:15 PM and 2:45 PM, what is the arc length traced by the tip of the minute hand?
    Answer: $8\pi$ cm From 2:15 PM to 2:45 PM is 30 minutes. In 30 minutes, the minute hand rotates through $\frac{30}{60} = \frac{1}{2}$ of a complete revolution. A complete revolution is $2\pi$ radians, so the minute hand rotates through $\frac{1}{2} \times 2\pi = \pi$ radians. Arc length = radius × angle = $8 \times \pi = 8\pi$ cm. Choice A uses radius 4 instead of 8. Choice C corresponds to 45 minutes of rotation. Choice D corresponds to a full hour of rotation.
13. The hands of a clock form an angle of 135° at a certain time. What is this angle in radians, and how many such angles would fit in one complete revolution?
    Answer: $\frac{3\pi}{4}$ radians, $2\frac{2}{3}$ angles Convert 135° to radians: $135° \cdot \frac{\pi}{180°} = \frac{3\pi}{4}$ radians. To find how many such angles fit in one complete revolution ($2\pi$ radians): $\frac{2\pi}{\frac{3\pi}{4}} = \frac{8\pi}{3\pi} = \frac{8}{3} = 2\frac{2}{3}$ angles.
14. The minute hand of a clock moves through an angle of 150° in a certain time period. If the minute hand is 6 cm long, what is the arc length traveled by the tip of the minute hand, expressed in terms of $\pi$?
    Answer: $5\pi$ cm First convert 150° to radians: $150° \cdot \frac{\pi}{180°} = \frac{150\pi}{180} = \frac{5\pi}{6}$ radians. Using $s = r\theta$: $s = 6 \cdot \frac{5\pi}{6} = 5\pi$ cm. Choice B uses radius 3 instead of 6. Choice C uses the degree measure directly with $\pi$. Choice D assumes $\theta = \frac{\pi}{2}$ radians (90°).
15. On a unit circle, a point moves from position $(1, 0)$ through an arc length of $\frac{7\pi}{3}$. After reducing to the equivalent position between $0$ and $2\pi$ radians, what angle in degrees corresponds to the final position?
    Answer: $60°$ This question tests your understanding of angular motion on the unit circle and the concept of coterminal angles. When a point moves through an arc length greater than one full revolution ($2\pi$ radians), you need to find the equivalent position within one standard rotation. Starting from $(1,0)$ and moving through $\frac{7\pi}{3}$ radians, you first need to reduce this angle to its equivalent position between $0$ and $2\pi$. Since one full revolution is $2\pi = \frac{6\pi}{3}$, you can write: $\frac{7\pi}{3} = \frac{6\pi}{3} + \frac{\pi}{3} = 2\pi + \frac{\pi}{3}$. This means the point completes one full revolution and then moves an additional $\frac{\pi}{3}$ radians. The final position is at $\frac{\pi}{3}$ radians. Converting to degrees: $\frac{\pi}{3} \times \frac{180°}{\pi} = 60°$. This confirms answer D is correct. Looking at the wrong answers: A) $420°$ represents the original $\frac{7\pi}{3}$ radians converted directly to degrees without reducing ($\frac{7\pi}{3} \times \frac{180°}{\pi} = 420°$). B) $240°$ equals $\frac{4\pi}{3}$ radians, which might come from incorrectly subtracting $\frac{7\pi}{3} - 2\pi = \frac{\pi}{3}$ but then using the wrong conversion. C) $300°$ equals $\frac{5\pi}{3}$ radians, possibly from calculation errors in the reduction process. Remember: when dealing with angles greater than $2\pi$ (or $360°$), always reduce by subtracting full rotations first, then convert units if needed. This two-step approach prevents common calculation errors.
16. Two concentric circles have radii of 4 cm and 7 cm. If a central angle intercepts arcs of combined length $11\pi$ cm on both circles, what is the measure of the central angle in radians?
    Answer: $\pi$ radians For the same central angle $\theta$, arc lengths are $s_1 = 4\theta$ and $s_2 = 7\theta$. Combined: $4\theta + 7\theta = 11\theta = 11\pi$, so $\theta = \pi$ radians. Choice B treats $11\pi$ as the arc length on one circle with radius 2. Choice C assumes the angle equals the total arc length divided by average radius. Choice D forgets the $\pi$ in the final answer.
17. A rotating wheel makes $\frac{5}{6}$ of a complete revolution. If the arc length traced by a point on the rim is $10\pi$ inches, what is the measure of the angle of rotation in degrees?
    Answer: 300° First find the radius: $\frac{5}{6}$ revolution = $\frac{5}{6} \cdot 2\pi = \frac{5\pi}{3}$ radians. Using $s = r\theta$: $10\pi = r \cdot \frac{5\pi}{3}$, so $r = 6$ inches. The angle in radians is $\frac{5\pi}{3}$, which converts to $\frac{5\pi}{3} \cdot \frac{180°}{\pi} = 300°$. Choice A uses $\frac{5}{6} \cdot 180° = 150°$ (incorrect fraction of semicircle). Choice C incorrectly uses $\frac{5}{6} \cdot 360° \cdot \frac{3}{5} = 216°$. Choice D assumes half revolution.
18. An angle measuring $-\frac{2\pi}{3}$ radians is coterminal with which of the following positive angles in degrees?
    Answer: 240° First convert $-\frac{2\pi}{3}$ to degrees: $-\frac{2\pi}{3} \cdot \frac{180°}{\pi} = -120°$. To find a positive coterminal angle, add $360°$: $-120° + 360° = 240°$. Choice B adds $420°$ instead of $360°$. Choice C gives the positive value of $-120°$ without adding $360°$. Choice D uses an incorrect conversion factor.

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