Radian Measurement of Angles          
\(x^2\)

Review Required Skills
Parts of Circle:

Term

Diameter=d  ,  Circumference  =  C  ,  Area  =  A  ,  Angle for full circle  =  360°

Formula  for Circle of radius, r

1. d= 2r
2. C=πd=2πr
3. A=πr²

Example : Quick Review of Required Skills

Consider circle with radius r where the minor arc a subtends an angle θ at the center O, as shown in the figure, for the following examples.

Q1. Find circumference for a circle with area of 16π m² 
Solution : Calculate radius from given area and the find the circumference.
A=πr²⇒16π=πr²⇒r=4
C=πd=2πr=2π(4)=8π
Circumference of the circle is 8π m.

Q2. Find area of the sector for r=10,θ=60°
Solution : Area of the circle=πr²
Since the sector angle is 60° and full circle angle is 360°, sector area will be one-sixth of that of the circle area.

Q3. Find length of the arc, a, for r=8 m,θ=120°
Solution : Circumference of a circle=2πr
Since angle subtended by the arc is 120° and full circle angle is 360°, arc length will be one-third of that of the circumference.

Radian

Radian is the standard unit for measuring angles.
Its symbol is C. Angle measurement of 2 radians can be written as 2 radians, 2², 2 rad or even just the number 2. The word ‘radian’ is an abbreviation of ‘radial’ angle.

In the figure, a is the arc length that subtends and angle θ at the center of a circle with radius r. For a given angle θ, the ratio of arc length to radius is always constant. This ratio represents the angle in radians.

The central angle  is one radian when arc length a is equal to the length of the radius r.
For a unit circle the length of an arc is numerically equal to the radian measure of the angle .
Alternate Proof for θ =

In a circle angle θ subtended is proportional to the arc length

                               (1)                                                       

If a²=r then θ₂= 1 radian

substituting we get

                                    (2)

Example: Subtended angle from radius and arc length

	Question	Solution
a.	Find the measure of central angle that is subtended by an arc length of 10 cm in a circle with radius 4 cm.
b.	Find the measure of central angle that is subtended by an arc length of 3 cm in a unit circle. NOTE: Unit circle has radius of 1	Angle subtended is numerically equal to the arc length

Example : Subtended angle from radius and arc length

	Question	Solution
a.	Find arc length of a circle with radius of 10 cm if central angle is 2c.
b.	Find the radius of the circle where arc length of 6 cm subtends an angle of 2 radians at the center.

Example 4: Angle subtended by a full circle
Show that the angle subtended by one rotation is
Solution 4: Angle subtended by a full circle

Example 5: Find arc length for given angle in a unit circle

Major arc on a unit circle subtends an angle of 5π/3 . Find exact length of the arc.

Solution 5: Find arc length for given angle in a unit circle

Radian – Degree Conversion

One full revolution of a point on the circle will subtend and angle of 360° or 2π radians.
These angles are equal. Therefore, 360°=2π radians ⇒180°=π radians

Example 1 : Degree – Radian Relation

	Question	Answer
a.	Find measure if 1 degree in radians	$$ 1 degree=\frac{π}{180°}  radians= 0.017453  radians$$
b.	Find the measure of 1 radian in degree	$$1 radians=\frac{π}{180°} = 57.297°=57.3°$$
NOTE: Equilateral triangle of side length r, as shown in the figure illustration that 1c is slightly less than 60°.

Table 1: Degree and radian measures of some common angles

Practice Q1: Convert from degrees to radians. Write exact and decimal equivalent.

a. 30°
b.135°
c.-90°
d.200°
e.-250°
Answers Q1

$$a. 30°=\frac{π}{6}=0.5236^c$$
$$b. 135°=\frac{3π}{4}=2.3562^c$$
$$c. -90°= \frac{-π}{2}=-1.5708^c$$
$$d. 200°=\frac{10}{9} rad=3.4906^c$$
$$r. -250°=\frac{-25π}{18}=4.3633^3$$

Note: When writing radian measure as fractional multiple of  unit ^c is normally not required. Radian measure in terms of  is more meaningful and is preferred over its decimal equivalent.

Practice Q2 : Convert each angle in radian to degrees. Show all the steps. Round to 2 decimal places, if required.
$$\frac{π}{3} radian$$
$$\frac{3π}{2} radian$$
$$\frac{-5π}{6} radian$$
$$3 radians$$
Answers Q2
$$\frac{π}{3} radian=60°$$
$$\frac{3π}{2} radian\270°$$
$$\frac{-5π}{6} radian=-150°$$
$$3 radians=171.89°$$
Note: Unit of radians is normally dropped when angle measure is given in faction of π.

Solution to Student’s Activity 1

Table 1: Convert the rotations around a circle into angles in radians and degrees.

Sketch Angles with Radian Measurement

Protractors are in degrees. So, how do we draw angles given in radians?
We observed from the Table 1 that the angles in radians can be represented as factions of a revolution. One full revolution represents angle of 2π radians and half of the revolution is π radians. To draw any angle in radians we need to divide the circle in the required denomination as illustrated in the examples.

\begin{document}
\hspace*{-3cm}
$$ {Example 8: Steps to Sketch Angle of  } \frac{4π}{3} $$
\end{document}
$${To sketch an angle of }\frac{4π}{3}{we need 4 angles of}\frac{π}{3}$$

$${πrepresent the angle of semi-circle (a straight line).} \ { Divide semi-circle in three equal parts (or circle in 6 equal parts) to get angle of }\frac{π}{3}\ { each.}\frac{4π}{3} \{ is the fourth} \frac{π}{3}{as shown in the figure.}$$

Convert Decimal to Degree(DMS), Minute, Seconds Form

1 degree =60 minutes
1 minute =60 seconds
1 degree =3600 seconds