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Angles and units of angular measurements

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Angles and units of angular measurements

  1. 1. Angles & units of angular measurements by SBR www.harekrishnahub.com
  2. 2. www.harekrishnahub.com Consider a line 𝑢𝑨. Let the line 𝑢𝑨 rotate about one of its endpoints, say 𝑢. Let the new position be 𝑢𝑨’. An angle is said to be generated. 𝑢𝑨 is the initial position and 𝑢𝑨’ is the final position. Therefore, an angle is the measure of the amount of rotation of a line from its initial position to the final position. An angle is denoted by βˆ π΄π‘‚π΄. Here, the point of rotation 𝑢 is called the vertex. The initial and final positions are called the initial and terminal sides respectively. They form the arms of the angle.
  3. 3. www.harekrishnahub.com If the line 𝑢𝑨 rotates in the anticlockwise direction, we say that the angle is positive. If the line 𝑢𝑨 rotates in the clockwise direction, we say that the angle is negative.
  4. 4. www.harekrishnahub.com Consider a line 𝑢𝑨 rotating about the origin 𝑢. Let it start from the position 𝑢𝑿 and rotate through an angle 𝜽 (anticlockwise) and occupy a position as shown below. Then βˆ π‘‹π‘‚π΄ = πœƒ The rotating line 𝑢𝑨 is called the radius vector. The positions 𝑢𝑿 and 𝑢𝑨 are called initial and terminal positions. If the rotating line 𝑢𝑨 coincides with 𝑢𝑿, 𝑢𝒀, 𝑢𝑿’ and 𝑢𝒀’, we get 𝟎°, πŸ—πŸŽΒ°, πŸπŸ–πŸŽΒ° and πŸπŸ•πŸŽΒ°. After rotating once, if OA coincides with the 𝒙 βˆ’ π’‚π’™π’Šπ’”, then 𝜽 = πŸ‘πŸ”πŸŽΒ°. The angles 𝟎°, πŸ—πŸŽΒ°, πŸπŸ–πŸŽΒ°, πŸπŸ•πŸŽΒ° and πŸ‘πŸ”πŸŽΒ° are called quadrant angles.
  5. 5. www.harekrishnahub.com If 𝑢𝑨 lies in 𝑰 quadrant, then we have, 𝟎 < 𝜽 < πŸ—πŸŽΒ° If 𝑢𝑨 lies in II quadrant, then we have, πŸ—πŸŽΒ° < 𝜽 < πŸπŸ–πŸŽΒ° If 𝑢𝑨 lies in 𝑰𝑰𝑰 quadrant, then we have, πŸπŸ–πŸŽΒ° < 𝜽 < πŸπŸ•πŸŽΒ° If 𝑢𝑨 lies in 𝑰𝑽 quadrant, then we have, πŸπŸ•πŸŽΒ° < 𝜽 < πŸ‘πŸ”πŸŽΒ° Let 𝑢𝑨 rotates 𝒏 times (counterclockwise) and comes back to its position, then the angle is taken as (𝒏 βœ• πŸ‘πŸ”πŸŽΒ° + 𝜽)
  6. 6. www.harekrishnahub.com Units of Angular Measurement 1. Sexagesimal measure. 2. Centesimal measure. 3. Radian (or circular) measure.
  7. 7. www.harekrishnahub.com Sexagesimal measure β€’ In this units of measurement, an angle is expressed in terms of degrees, minutes and seconds. β€’ Consider a horizontal line and a vertical line. The angle between them is a right angle. The right angle is divided into 90 equal parts. Each part is called a degree denoted by (⁰). Each degree is in turn divided into 60 equal parts and each part is called a minute denoted by (β€˜). Each minute is in turn divided into 60 equal parts and each part is called a second denoted by (β€œ). β€’ Therefore, we have 𝟏 π‘Ήπ’Šπ’ˆπ’‰π’• π’‚π’π’ˆπ’π’† = πŸ—πŸŽ 𝒐 𝟏 𝒐 = πŸ”πŸŽβ€² πŸβ€² = πŸ”πŸŽ" In this unit, an angle is expressed as: 𝑿 degrees, 𝒀 minutes, 𝒁 seconds. For example: πŸ•πŸ’ 𝒐 πŸ’πŸ–β€² πŸ‘πŸ“"
  8. 8. www.harekrishnahub.com Centesimal measure β€’ In this system, a right angle is divided into 100 equal parts. β€’ Each part is called a grade. β€’ Each grade is further divided into 100 equal parts and each part is called a minute. β€’ Each minute is in turn divided into 100 equal parts and each such part is called a second. 𝟏 π‘Ήπ’Šπ’ˆπ’‰π’• π’‚π’π’ˆπ’π’† = 𝟏𝟎𝟎 π’ˆπ’“π’‚π’…π’†π’” = 𝟏𝟎𝟎 π’ˆ 𝟏 π’ˆ = πŸπŸŽπŸŽβ€² πŸβ€² = πŸπŸŽπŸŽβ€œ πŸ—πŸŽ 𝒐 = 𝟏𝟎𝟎 π’ˆ 𝟏 𝒐 = 𝟏𝟎𝟎 πŸ—πŸŽ π’ˆ = 𝟏𝟎 πŸ— π’ˆ 𝟏 π’ˆ = πŸ— 𝟏𝟎 𝒐
  9. 9. www.harekrishnahub.com Radian or Circular measure Consider a circle with centre 𝑢 and radius 𝒓 units. Let 𝑨 and 𝑩 be any points on the circle, such that the length of the arc 𝑨𝑩 is equal to the radius 𝒓 of the circle. That is, arc 𝐴𝐡 = π‘Ÿ. Join 𝑢𝑨 and 𝑢𝑩. Then the angle βˆ π΄π‘‚π΅ will be equal to 𝟏 π’“π’‚π’…π’Šπ’‚π’ or 𝟏 π’„π’Šπ’“π’„π’–π’π’‚π’“ π’Žπ’†π’‚π’”π’–π’“π’†. Therefore, a radian is defined as the angle subtended at the centre of the circle by an arc of length equal to the radius of the circle.

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