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Introduction

• Circular Measure:
  • Refers to the measurement of angles using radians instead of degrees.
  • Radian is considered a “natural unit” of angular measure and simplifies many mathematical formulas and calculations.
• Definition of a Radian:
  • A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
  • Key relationship: 2π radians=360∘2pi , text{radians} = 360^circ

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Key Relationships Between Degrees and Radians

1. Conversion Between Degrees and Radians

• Conversion formula:Radians=Degrees×π180text{Radians} = text{Degrees} times frac{pi}{180} Degrees=Radians×180πtext{Degrees} = text{Radians} times frac{180}{pi}
• Worked example:
  • Convert 60∘60^circ to radians:Radians=60×π180=π3text{Radians} = 60 times frac{pi}{180} = frac{pi}{3}
  • Convert π4frac{pi}{4} radians to degrees:Degrees=π4×180π=45∘text{Degrees} = frac{pi}{4} times frac{180}{pi} = 45^circ

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Applications of Radians

1. Arc Length

• Formula:Arc Length=rθtext{Arc Length} = rthetawhere:
  • rr: Radius of the circle.
  • θtheta: Angle in radians.
• Worked example:
  • A circle has a radius of 8 cm, and an angle of π3frac{pi}{3} radians subtends an arc. Find the arc length: Arc Length=8×π3=8π3 cm.text{Arc Length} = 8 times frac{pi}{3} = frac{8pi}{3} , text{cm}.

2. Sector Area

• Formula:Sector Area=12r2θtext{Sector Area} = frac{1}{2} r^2 thetawhere:
  • rr: Radius.
  • θtheta: Angle in radians.
• Worked example:
  • A sector has a radius of 6 cm and an angle of 22 radians. Find the area: Sector Area=12×62×2=36 cm2.text{Sector Area} = frac{1}{2} times 6^2 times 2 = 36 , text{cm}^2.

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Understanding the Geometry

1. Comparison Between Degrees and Radians

• Angles in radians provide a proportional relationship between the arc length and the radius.
• This simplifies trigonometric calculations in advanced mathematics.

2. Practical Example

• In a circle with radius r=10r = 10:
  • For θ=π/2theta = pi/2 radians (90°), the arc length is: Arc Length=10×π2=5π.text{Arc Length} = 10 times frac{pi}{2} = 5pi.
  • For θ=πtheta = pi radians (180°), the arc length is: Arc Length=10×π=10π.text{Arc Length} = 10 times pi = 10pi.

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Worked Examples

Example 1: Finding the Angle in Radians

• A sector has a radius of 7 cm and an arc length of 11 cm. Find the angle subtended by the arc at the center.
• Solution:
  • Use the formula: θ=Arc Lengthrtheta = frac{text{Arc Length}}{r}
  • Substituting: θ=117 radians.theta = frac{11}{7} , text{radians}.

Example 2: Perimeter of a Sector

• A sector of a circle has a radius of 4 cm and an angle of 1.51.5 radians. Find the perimeter of the sector.
• Solution:
  • Perimeter: Perimeter=2r+Arc Lengthtext{Perimeter} = 2r + text{Arc Length}
  • Arc length: Arc Length=rθ=4×1.5=6 cm.text{Arc Length} = rtheta = 4 times 1.5 = 6 , text{cm}.
  • Total perimeter: Perimeter=2×4+6=14 cm.text{Perimeter} = 2 times 4 + 6 = 14 , text{cm}.

Example 3: Area of a Shaded Segment

• A circle has a radius of 5 cm, and a sector subtends an angle of π3frac{pi}{3} radians. Find the area of the segment formed by the sector.
• Solution:
  • Area of the sector: Sector Area=12r2θ=12×52×π3=25π6 cm2.text{Sector Area} = frac{1}{2} r^2 theta = frac{1}{2} times 5^2 times frac{pi}{3} = frac{25pi}{6} , text{cm}^2.
  • Area of the triangle: Triangle Area=12r2sin⁡θ=12×52×sin⁡(π3)=2538.text{Triangle Area} = frac{1}{2} r^2 sintheta = frac{1}{2} times 5^2 times sinleft(frac{pi}{3}right) = frac{25sqrt{3}}{8}.
  • Area of the segment: Segment Area=Sector Area−Triangle Area.text{Segment Area} = text{Sector Area} – text{Triangle Area}.

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Practice Problems

1. Convert the following to radians:
   • 45∘,120∘,270∘45^circ, 120^circ, 270^circ.
2. Find the arc length of a circle with a radius of 10 cm and an angle of 2π3frac{2pi}{3} radians.
3. Determine the sector area of a circle with radius 8 cm and angle 1.21.2 radians.
4. Calculate the perimeter of a sector with radius 7 cm and an angle of 2.52.5 radians.

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Summary

• Radians: A natural unit for measuring angles, directly relating arc length to radius.
• Conversions: Easy transitions between degrees and radians simplify trigonometric functions.
• Formulas:
  • Arc length: Arc Length=rθtext{Arc Length} = rtheta
  • Sector area: Sector Area=12r2θtext{Sector Area} = frac{1}{2} r^2 theta
• Applications range from geometry to advanced mathematical modeling.

Key Concepts

Arc of a Circle

• Definition:
  • An arc is a segment of the circumference of a circle.
  • The length of an arc depends on the radius of the circle and the angle subtended at the center.

Formula for Arc Length

• For a circle with radius rr, and an angle θtheta measured in radians, the arc length (LL) is given by: L=rθL = r theta
  • rr: Radius of the circle.
  • θtheta: Angle subtended at the center (in radians).

Why Radians?

• Radians directly relate the arc length to the radius, making calculations simpler compared to degrees.
• Conversion from degrees to radians: Radians=Degrees×π180text{Radians} = text{Degrees} times frac{pi}{180}

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Detailed Derivation of Arc Length Formula

1. Radian Definition:
   • A radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
   • Arc Length=rtext{Arc Length} = r when θ=1 radiantheta = 1 , text{radian}.
2. Proportional Relationship:
   • For any angle θtheta (in radians): Arc Length=rθtext{Arc Length} = r theta
3. Using Degrees:
   • When θtheta is in degrees, convert to radians first: Arc Length=r(θ×π180)text{Arc Length} = r left(theta times frac{pi}{180}right)

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Applications of the Formula

1. Finding Arc Length

• Given radius and angle, calculate the arc length directly.

2. Finding Radius

• Rearrange the formula to find the radius: r=Arc Lengthθr = frac{text{Arc Length}}{theta}

3. Finding Angle

• Rearrange the formula to find the angle in radians: θ=Arc Lengthrtheta = frac{text{Arc Length}}{r}

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Worked Examples

Example 1: Arc Length

Find the arc length of a circle with radius 8 cm and an angle of π3frac{pi}{3} radians.

• Solution: L=rθ=8×π3=8π3 cmL = r theta = 8 times frac{pi}{3} = frac{8pi}{3} , text{cm}

Example 2: Radius

A sector has an arc length of 9.6 cm and an angle of 2 radians. Find the radius of the circle.

• Solution: r=Arc Lengthθ=9.62=4.8 cmr = frac{text{Arc Length}}{theta} = frac{9.6}{2} = 4.8 , text{cm}

Example 3: Angle

A circle has a radius of 5 cm and an arc length of 12 cm. Find the angle subtended by the arc at the center.

• Solution: θ=Arc Lengthr=125=2.4 radianstheta = frac{text{Arc Length}}{r} = frac{12}{5} = 2.4 , text{radians}

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Special Cases

1. Complete Circle

• The angle for a complete circle is 2π2pi radians.
• Total arc length (circumference): L=r×2π=2πrL = r times 2pi = 2pi r

2. Half Circle (Semicircle)

• The angle for a semicircle is πpi radians.
• Arc length: L=rπL = r pi

3. Quarter Circle

• The angle for a quarter circle is π2frac{pi}{2} radians.
• Arc length: L=rπ2L = r frac{pi}{2}

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Perimeter of a Sector

• The perimeter of a circular sector includes the arc and two radii: Perimeter=2r+Ltext{Perimeter} = 2r + L where L=rθL = r theta.

Example:

• A sector of a circle has radius 6 cm and angle 1.21.2 radians. Find the perimeter.
  • Arc Length: L=rθ=6×1.2=7.2 cmL = r theta = 6 times 1.2 = 7.2 , text{cm}
  • Perimeter: Perimeter=2r+L=2(6)+7.2=19.2 cmtext{Perimeter} = 2r + L = 2(6) + 7.2 = 19.2 , text{cm}

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Practice Problems

1. Arc Length:
   • Find the arc length of a circle with:
     • Radius: 10 cm, Angle: π4frac{pi}{4} radians.
   • Solution: L=10×π4=10π4=5π2 cmL = 10 times frac{pi}{4} = frac{10pi}{4} = frac{5pi}{2} , text{cm}
2. Finding Radius:
   • Arc Length: 14 cm, Angle: π3frac{pi}{3} radians.
   • Solution: r=14π3=42π cmr = frac{14}{frac{pi}{3}} = frac{42}{pi} , text{cm}
3. Finding Angle:
   • Radius: 9 cm, Arc Length: 15 cm.
   • Solution: θ=159=1.67 radianstheta = frac{15}{9} = 1.67 , text{radians}

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Key Observations

• Always ensure the angle is in radians when using the formula L=rθL = r theta.
• For calculations in degrees, convert to radians using: θ(radians)=θ(degrees)×π180theta (text{radians}) = theta (text{degrees}) times frac{pi}{180}

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Applications in Real Life

• Engineering: Calculating distances on curved paths.
• Astronomy: Measuring celestial arcs.
• Physics: Analyzing motion along curved trajectories.

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Summary

• Arc length is a fundamental concept in circular geometry, directly linking radius, angle, and arc.
• The simplicity of the formula L=rθL = r theta makes it versatile for various mathematical and real-world problems.

Introduction

• The area of a sector is a portion of the area of a circle, corresponding to a specific angle at the center.
• Sectors are defined by:
  • Radius (rr).
  • Angle subtended at the center (θtheta).
• The area of a sector is derived using the proportional relationship between the angle of the sector and the total angle of a circle.

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Formula for the Area of a Sector

1. Basic Proportionality:
   • The area of a sector is proportional to the angle subtended at the center.
   • For a full circle: Area of Circle=πr2,Central Angle=2π radians.text{Area of Circle} = pi r^2, quad text{Central Angle} = 2pi , text{radians}.
   • For a sector with an angle θtheta (in radians): Area of Sector=θ2π×πr2.text{Area of Sector} = frac{theta}{2pi} times pi r^2.
2. Simplified Formula:
   • When θtheta is in radians: Area of Sector=12r2θ.text{Area of Sector} = frac{1}{2} r^2 theta.

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Deriving the Formula

1. Start with the general proportionality:Area of Sector=θ2π×πr2.text{Area of Sector} = frac{theta}{2pi} times pi r^2.Simplify:Area of Sector=θ2×r2=12r2θ.text{Area of Sector} = frac{theta}{2} times r^2 = frac{1}{2} r^2 theta.
2. This formula applies only if θtheta is measured in radians. If θtheta is in degrees, convert it to radians first:θ(radians)=θ(degrees)×π180.theta (text{radians}) = theta (text{degrees}) times frac{pi}{180}.

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Worked Examples

Example 1: Calculating the Area

Find the area of a sector with:

• Radius: 6 cm.
• Angle: 1.5 radians1.5 , text{radians}.
• Solution: Use the formula:Area=12r2θ.text{Area} = frac{1}{2} r^2 theta.Substitute:Area=12×62×1.5=27 cm2.text{Area} = frac{1}{2} times 6^2 times 1.5 = 27 , text{cm}^2.

Example 2: Determining the Angle

A sector has:

• Radius: 4 cm.
• Area: 10 cm210 , text{cm}^2. Find the angle subtended at the center.
• Solution: Rearrange the formula to solve for θtheta:θ=2×Arear2.theta = frac{2 times text{Area}}{r^2}.Substitute:θ=2×1042=2016=1.25 radians.theta = frac{2 times 10}{4^2} = frac{20}{16} = 1.25 , text{radians}.

Example 3: Finding the Radius

A sector has:

• Area: 15 cm215 , text{cm}^2.
• Angle: 2 radians2 , text{radians}. Find the radius.
• Solution: Rearrange the formula to solve for rr:r=2×Areaθ.r = sqrt{frac{2 times text{Area}}{theta}}.Substitute:r=2×152=15≈3.87 cm.r = sqrt{frac{2 times 15}{2}} = sqrt{15} approx 3.87 , text{cm}.

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Shaded Regions Involving Sectors

1. Area of a Segment:
   • A segment is the area of a sector minus the area of the triangle formed by the two radii and the arc.
   • Formula: Segment Area=Sector Area−Triangle Area.text{Segment Area} = text{Sector Area} – text{Triangle Area}.
   • Example:
     • For a sector with:
       • Radius: 8 cm8 , text{cm},
       • Angle: 2 radians2 , text{radians}:
       • Area of sector: Sector Area=12×82×2=64 cm2.text{Sector Area} = frac{1}{2} times 8^2 times 2 = 64 , text{cm}^2.
       • Area of triangle: Triangle Area=12×8×8×sin⁡(2)≈29.1 cm2.text{Triangle Area} = frac{1}{2} times 8 times 8 times sin(2) approx 29.1 , text{cm}^2.
       • Segment area: Segment Area=64−29.1=34.9 cm2.text{Segment Area} = 64 – 29.1 = 34.9 , text{cm}^2.
2. Perimeter of a Sector:
   • Perimeter includes the arc length and two radii: Perimeter=2r+Arc Length.text{Perimeter} = 2r + text{Arc Length}.
   • Arc length: Arc Length=rθ.text{Arc Length} = rtheta.

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Applications

1. Geometry Problems:
   • Sectors are used to calculate areas of pie-shaped regions in circles.
2. Engineering:
   • Determining coverage areas in radar and communication sectors.
3. Physics:
   • Used in circular motion to analyze angular displacement.

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Practice Problems

1. Find the area of a sector with:
   • Radius: 5 cm5 , text{cm},
   • Angle: π3 radians frac{pi}{3} , text{radians}.
   • Solution: Area=12×52×π3=25π6 cm2.text{Area} = frac{1}{2} times 5^2 times frac{pi}{3} = frac{25pi}{6} , text{cm}^2.
2. A sector has an area of 30 cm230 , text{cm}^2 and an angle of 1.5 radians1.5 , text{radians}. Find the radius.
   • Solution: r=2×301.5=40≈6.32 cm.r = sqrt{frac{2 times 30}{1.5}} = sqrt{40} approx 6.32 , text{cm}.
3. A sector with radius 7 cm7 , text{cm} subtends an angle of 2 radians2 , text{radians}. Find its perimeter.
   • Solution: Perimeter=2r+rθ=2×7+7×2=28 cm.text{Perimeter} = 2r + rtheta = 2 times 7 + 7 times 2 = 28 , text{cm}.

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Summary

• The area of a sector is derived using the proportionality between the sector angle and the full angle of a circle.
• Formula: Area of Sector=12r2θ (radians).text{Area of Sector} = frac{1}{2} r^2 theta , (text{radians}).
• Applications of sectors extend across geometry, physics, and engineering, making them essential in various fields.