Trigonometry- Arc Length

The definition of a radian meadure
An angle of a 1 radian
Examples!!



Let the letter s (for space) symbolize the length of an arc, which is called arc length.
Now the circumference of a circle is an arc length.
The ratio of the circumference to the diameter is the basis of radian measure.
That ratio is the definition of π.

π = C/D
C = Circumference
D = Diameter

Since D = 2r, then

π = C/2r
or,
2π = C/r




That ratio of the circumference of a circle C to the radius r -- 2π -- is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.




Radian measure = θ s /r

Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that indicates the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle θ.




EXAMPLE:
In a circle whose radius is 10 cm, a central angle θ intercepts an arc of 8 cm.



a)What is the radian measure of that angle?
b) what is the arc length if the radius is 5 cm?

Answers
a)
θ = s /r
= 8/ 10
= .8
b)For a given central angle, the ratio of arc to radius is the same. 5 is half of 10. Therefore the arc length will be half of 8: 4cm.
An angle of 1 radian
Note that an angle of 1 radian is a central angle whose subtending arc is equal in length to the radius.