Notes On Angles - Gujarat board Class 11 Maths
Consider two rays OA and OB originating from a point O. If one ray is fixed and the other is rotated about it, an angle is formed.   Angle An angle is a measure of rotation of a given ray about its initial point. Ray OA is called the initial side of the angle and the destination ray OB is called the terminal side of the angle.  If the direction of rotation is counter clockwise, then the magnitude of the angle is conventionally taken as positive.     Now, let ray OA be rotated in the clockwise direction to meet ray OB.   In this case, the magnitude of the angle is negative.  The point of origin of the two rays of an angle is called the vertex.     When the initial ray, OA, takes one complete revolution and comes back to its original position, then it makes an angle of three hundred and sixty degrees.   When rays OA and OB are in the opposite directions, an angle of one hundred and eighty degrees is formed.     Units of Measurement of an Angle The two most common units for the measurement of an angle are: 1. Degree measure 2. Radian measure Degree measure An angle is of measure one degree if the rotation from the initial side to the terminal side is $\frac{\text{1}}{\text{360}}$th of a revolution. 1 degree = 1° A degree is divided into 60 equal parts, and each part is called a minute. i.e. 1° = 60° (read as 60 minutes) Or $\text{}\frac{\text{1}}{\text{60}}⁰$ = 1’ A minute is further divided into 60 equal parts, and each part is called a second. i.e. 1’ = 60” (read as 60 seconds) Or $\frac{\text{1}}{\text{60}}$’ = 1” 45° 15’  = 45° + $\left(\mathrm{45}\frac{1}{4}\right)\text{⁰}$[ $\because \text{}\left(\frac{1}{60}\right)\text{⁰}\mathrm{\text{= 1'}}$ ] The figures depict some angles in degree measure.     Radian measure The angle subtended by an arc at the centre of a circle, whose length is equal to the radius of the circle is 1 radian. Let θ be the angle subtended by an arc of a circle of radius r. If the length of the arc is also r, then θ  = 1 radian. If the radius of the circle and the length of the arc are equal to 1 unit, then  θ  = 1 radian.   In a circle of radius r, an arc of length r subtends an angle = 1 radian. $\therefore$ An arc of length l subtends an angle = $\frac{\text{l}}{\text{r}}$ radians. Hence, in a circle of radius r, if an arc of length l subtends an angle θ radians at the centre, then l = rθ.  Relation between radian and degree   Consider a unit circle. Circumference of circle of radius 1 unit = 2π  One complete revolution of the initial side subtends an angle of 2π  radians.   One complete revolution of the initial side is equal to 360°.   ⇒ 360° = 2π radian ⇒ 180° = π radian The table shows the degree measures and radian measures of some standard angles.     Degree    30°     45°    60°    90°    180°    270°    360°   Radian     $\frac{\pi }{\text{6}}$     $\frac{\pi }{\text{4}}$    $\frac{\pi }{\text{3}}$    $\frac{\pi }{\text{2}}$     $\frac{}{\text{}}$π    $\frac{3\pi }{\text{2}}$     2π   1 radian =( $\frac{\text{180}}{\pi }$ )°= (180 × $\frac{\text{7}}{\text{22}}$ )° = 57° 16' (approx) 1° = $\frac{\pi }{\text{180}}$radian = 0.01746 radian  (approx) 1° = $\frac{\pi }{\text{180}}$radian or 1 radian = $\frac{\text{180}}{\pi }$ degrees.   Angles can be converted from degree measure to radian measure or vice versa using the formulae.   If D represents the degrees and R represents the radians, then $\frac{\text{D}}{\text{90}}$  = $\frac{\text{R}}{\left(\frac{\pi }{\text{2}}\right)}$

Summary

Consider two rays OA and OB originating from a point O. If one ray is fixed and the other is rotated about it, an angle is formed.   Angle An angle is a measure of rotation of a given ray about its initial point. Ray OA is called the initial side of the angle and the destination ray OB is called the terminal side of the angle.  If the direction of rotation is counter clockwise, then the magnitude of the angle is conventionally taken as positive.     Now, let ray OA be rotated in the clockwise direction to meet ray OB.   In this case, the magnitude of the angle is negative.  The point of origin of the two rays of an angle is called the vertex.     When the initial ray, OA, takes one complete revolution and comes back to its original position, then it makes an angle of three hundred and sixty degrees.   When rays OA and OB are in the opposite directions, an angle of one hundred and eighty degrees is formed.     Units of Measurement of an Angle The two most common units for the measurement of an angle are: 1. Degree measure 2. Radian measure Degree measure An angle is of measure one degree if the rotation from the initial side to the terminal side is $\frac{\text{1}}{\text{360}}$th of a revolution. 1 degree = 1° A degree is divided into 60 equal parts, and each part is called a minute. i.e. 1° = 60° (read as 60 minutes) Or $\text{}\frac{\text{1}}{\text{60}}⁰$ = 1’ A minute is further divided into 60 equal parts, and each part is called a second. i.e. 1’ = 60” (read as 60 seconds) Or $\frac{\text{1}}{\text{60}}$’ = 1” 45° 15’  = 45° + $\left(\mathrm{45}\frac{1}{4}\right)\text{⁰}$[ $\because \text{}\left(\frac{1}{60}\right)\text{⁰}\mathrm{\text{= 1'}}$ ] The figures depict some angles in degree measure.     Radian measure The angle subtended by an arc at the centre of a circle, whose length is equal to the radius of the circle is 1 radian. Let θ be the angle subtended by an arc of a circle of radius r. If the length of the arc is also r, then θ  = 1 radian. If the radius of the circle and the length of the arc are equal to 1 unit, then  θ  = 1 radian.   In a circle of radius r, an arc of length r subtends an angle = 1 radian. $\therefore$ An arc of length l subtends an angle = $\frac{\text{l}}{\text{r}}$ radians. Hence, in a circle of radius r, if an arc of length l subtends an angle θ radians at the centre, then l = rθ.  Relation between radian and degree   Consider a unit circle. Circumference of circle of radius 1 unit = 2π  One complete revolution of the initial side subtends an angle of 2π  radians.   One complete revolution of the initial side is equal to 360°.   ⇒ 360° = 2π radian ⇒ 180° = π radian The table shows the degree measures and radian measures of some standard angles.     Degree    30°     45°    60°    90°    180°    270°    360°   Radian     $\frac{\pi }{\text{6}}$     $\frac{\pi }{\text{4}}$    $\frac{\pi }{\text{3}}$    $\frac{\pi }{\text{2}}$     $\frac{}{\text{}}$π    $\frac{3\pi }{\text{2}}$     2π   1 radian =( $\frac{\text{180}}{\pi }$ )°= (180 × $\frac{\text{7}}{\text{22}}$ )° = 57° 16' (approx) 1° = $\frac{\pi }{\text{180}}$radian = 0.01746 radian  (approx) 1° = $\frac{\pi }{\text{180}}$radian or 1 radian = $\frac{\text{180}}{\pi }$ degrees.   Angles can be converted from degree measure to radian measure or vice versa using the formulae.   If D represents the degrees and R represents the radians, then $\frac{\text{D}}{\text{90}}$  = $\frac{\text{R}}{\left(\frac{\pi }{\text{2}}\right)}$