Angle
An angle is made up of two rays with a common end point. This end point is the vertex of the angle. The rays are the sides of the angle.
Angle in trigonometry
Let a revolving ray starting from OX, rotate about O in a plane and stop at position OP. Then it is said to describe an angle XOP.
OX is called initial side.
OP is called final/terminal side.
O, the point of rotation is the vertex of the angle.
If rotation is anticlockwise, the angle is positive.
If rotation is clockwise, the angle is negative.
The measure of an angle is the amount of rotation from the initial side to the terminal side.
One unit of measuring angle is, one complete rotation(revolution).
This unit is convenient only for large angles.
Therefore, most commonly used units of measurement are degree measure and radian measure.
Degree measure
In this system, an angle is measured in degrees, minutes and seconds.
A complete rotation describes 360° i.e. $\displaystyle \small 1^{\circ}=\frac{1}{360}th$ of a complete rotation.
∴ 1 right angle = 90° (since right angle is $\displaystyle \small \frac{1}{4}th$ of full rotation)
A degree is further subdivided as,
1 degree = 60 minutes, written as 1° = 60'
1 minute = 60 seconds, written as 1' = 60''
Radian measure
If a central angle θ, in a circle of radius r, cuts off an arc of length s, then the measure of θ in radians, is given by,
$\displaystyle \small \theta =\frac{s}{r}$
In a circle, a central angle that cuts off an arc equal in length to the radius of the circle, has a measure of 1 radian.
i.e.when s = r,
θ = 1 radian
Example: A central angle θ in a circle of radius 3cm cuts off an arc of length 6cm. What is the radian measure of θ?
$\displaystyle \small \theta =\frac{s}{r}=\frac{6}{3}$
θ = 2 radians
Relation between Degree and Radian
The angle formed by one full rotation about the center of a circle of radius r will cut off an arc equal to the circumference of the circle.
Since the circumference of a circle of radius r is 2Ï€r,
∴ $\displaystyle \small \theta =\frac{2\pi r}{r}=2\pi$ radians
θ measures one full rotation i.e. 360°
∴ 2Ï€ radians = 360°
Ï€ radians = 180°
Conversion Formulas
(i) Degree to Radian
Radian measure = $\displaystyle \small \frac{\pi }{180}$ × Degree measure
(ii) Radian to Degree
Degree measure = $\displaystyle \small \frac{180}{\pi }$ × Radian measure
Also, by substituting π = 22/7 = 3.14,
1 radian ≈ 57°16'
1 degree ≈ 0.01746 radian
Hints for Solving Problems
(i) Degree to Radians
If degree is given as d°m's'' [degree minutes seconds] format, first convert this into only degree.
degree = $\displaystyle \small d^{\circ}+\left ( \frac{m}{60} \right )^{\circ}+\left ( \frac{s}{3600} \right )^{\circ}$
Example: 10°39'17''
10°39'17'' = $\displaystyle \small 10^{\circ}+\left ( \frac{39}{60} \right )^{\circ}+\left ( \frac{17}{3600} \right )^{\circ}$
= 10°+0.65°+0.0047°
= 10.6547°
Now for this degree value multiply π/180
(ii) Radians to Degree
For the given radian value multiply 180/Ï€. Substitute Ï€ = 22/7. Convert the result to d°m's''
Example: If the result after multiplying 180/Ï€ is, $\displaystyle \small 343\frac{7}{11}$
$\displaystyle \small 343\frac{7}{11}$ = 343° + $\displaystyle \small \left ( \frac{7}{11} \right )*60$ minute
= 343° + 38$\displaystyle \small \frac{2}{11}$ minute
= 343° + 38' +$\displaystyle \small \left ( \frac{2}{11} \right )*60$ second
= 343° + 38' + 10.9'' [take approximate value for seconds]
= 343°38'11''
An angle is made up of two rays with a common end point. This end point is the vertex of the angle. The rays are the sides of the angle.
Angle in trigonometry
OX is called initial side.
OP is called final/terminal side.
O, the point of rotation is the vertex of the angle.
If rotation is anticlockwise, the angle is positive.
If rotation is clockwise, the angle is negative.
The measure of an angle is the amount of rotation from the initial side to the terminal side.
One unit of measuring angle is, one complete rotation(revolution).
Therefore, most commonly used units of measurement are degree measure and radian measure.
Degree measure
In this system, an angle is measured in degrees, minutes and seconds.
A complete rotation describes 360° i.e. $\displaystyle \small 1^{\circ}=\frac{1}{360}th$ of a complete rotation.
∴ 1 right angle = 90° (since right angle is $\displaystyle \small \frac{1}{4}th$ of full rotation)
A degree is further subdivided as,
1 degree = 60 minutes, written as 1° = 60'
1 minute = 60 seconds, written as 1' = 60''
$\displaystyle \small \theta =\frac{s}{r}$
In a circle, a central angle that cuts off an arc equal in length to the radius of the circle, has a measure of 1 radian.
i.e.when s = r,
θ = 1 radian
Example: A central angle θ in a circle of radius 3cm cuts off an arc of length 6cm. What is the radian measure of θ?
$\displaystyle \small \theta =\frac{s}{r}=\frac{6}{3}$
θ = 2 radians
Relation between Degree and Radian
The angle formed by one full rotation about the center of a circle of radius r will cut off an arc equal to the circumference of the circle.
Since the circumference of a circle of radius r is 2Ï€r,
∴ $\displaystyle \small \theta =\frac{2\pi r}{r}=2\pi$ radians
θ measures one full rotation i.e. 360°
∴ 2Ï€ radians = 360°
Ï€ radians = 180°
| Degree | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|
| Radian | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
Conversion Formulas
(i) Degree to Radian
Radian measure = $\displaystyle \small \frac{\pi }{180}$ × Degree measure
(ii) Radian to Degree
Degree measure = $\displaystyle \small \frac{180}{\pi }$ × Radian measure
Also, by substituting π = 22/7 = 3.14,
1 radian ≈ 57°16'
1 degree ≈ 0.01746 radian
Hints for Solving Problems
(i) Degree to Radians
If degree is given as d°m's'' [degree minutes seconds] format, first convert this into only degree.
degree = $\displaystyle \small d^{\circ}+\left ( \frac{m}{60} \right )^{\circ}+\left ( \frac{s}{3600} \right )^{\circ}$
Example: 10°39'17''
10°39'17'' = $\displaystyle \small 10^{\circ}+\left ( \frac{39}{60} \right )^{\circ}+\left ( \frac{17}{3600} \right )^{\circ}$
= 10°+0.65°+0.0047°
= 10.6547°
Now for this degree value multiply π/180
(ii) Radians to Degree
For the given radian value multiply 180/Ï€. Substitute Ï€ = 22/7. Convert the result to d°m's''
Example: If the result after multiplying 180/Ï€ is, $\displaystyle \small 343\frac{7}{11}$
$\displaystyle \small 343\frac{7}{11}$ = 343° + $\displaystyle \small \left ( \frac{7}{11} \right )*60$ minute
= 343° + 38$\displaystyle \small \frac{2}{11}$ minute
= 343° + 38' +$\displaystyle \small \left ( \frac{2}{11} \right )*60$ second
= 343° + 38' + 10.9'' [take approximate value for seconds]
= 343°38'11''
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