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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 πΆπ¨. Let the line πΆπ¨ rotate about one of its endpoints, say πΆ. Let the new position be πΆπ¨β. An angle is said to be generated. πΆπ¨ is the initial position and πΆπ¨β 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 β π΄ππ΄. Here, the point of rotation πΆ 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 πΆπ¨ rotates in the anticlockwise direction, we say that the angle is positive. If the line πΆπ¨ rotates in the clockwise direction, we say that the angle is negative.
4. 4. www.harekrishnahub.com Consider a line πΆπ¨ rotating about the origin πΆ. Let it start from the position πΆπΏ and rotate through an angle π½ (anticlockwise) and occupy a position as shown below. Then β πππ΄ = π The rotating line πΆπ¨ is called the radius vector. The positions πΆπΏ and πΆπ¨ are called initial and terminal positions. If the rotating line πΆπ¨ coincides with πΆπΏ, πΆπ, πΆπΏβ and πΆπβ, we get πΒ°, ππΒ°, πππΒ° and πππΒ°. After rotating once, if OA coincides with the π β ππππ, then π½ = πππΒ°. The angles πΒ°, ππΒ°, πππΒ°, πππΒ° and πππΒ° are called quadrant angles.
5. 5. www.harekrishnahub.com If πΆπ¨ lies in π° quadrant, then we have, π οΌ π½ οΌ ππΒ° If πΆπ¨ lies in II quadrant, then we have, ππΒ° οΌ π½ οΌ πππΒ° If πΆπ¨ lies in π°π°π° quadrant, then we have, πππΒ° οΌ π½ οΌ πππΒ° If πΆπ¨ lies in π°π½ quadrant, then we have, πππΒ° οΌ π½ οΌ πππΒ° Let πΆπ¨ rotates π times (counterclockwise) and comes back to its position, then the angle is taken as (π β πππΒ° + π½)
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 (β°). 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). β’ Therefore, we have π πΉππππ πππππ = ππ π π π = ππβ² πβ² = ππ" In this unit, an angle is expressed as: πΏ degrees, π minutes, π seconds. For example: ππ π ππβ² ππ"
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. π πΉππππ πππππ = πππ ππππππ = πππ π π π = πππβ² πβ² = πππβ ππ π = πππ π π π = πππ ππ π = ππ π π π π = π ππ π
9. 9. www.harekrishnahub.com Radian or Circular measure Consider a circle with centre πΆ and radius π units. Let π¨ and π© be any points on the circle, such that the length of the arc π¨π© is equal to the radius π of the circle. That is, arc π΄π΅ = π. Join πΆπ¨ and πΆπ©. Then the angle β π΄ππ΅ will be equal to π ππππππ or π ππππππππ ππ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.