Trigonometry
     Trigonometric Functions
      Unit of measurement of angles - a radian (the circular measure)
         Protractor - an instrument for measuring angles
         Degrees to radians and radians to degrees conversion examples
Unit of measurement of angles - a radian (the circular measure)
The relation between a central angle a (the angle between two radii) and the corresponding arc l in the circle of radius r is shown by the proportion,
a : 360 = l : 2rp
It shows that the central angle a compared to the round angle of 360
(called perigon) is in the same relation as the corresponding arc l 
compared to the circumference 2rp. Therefore,
 
where the ratio we call the circular measure, usually denoted arad, i.e.,
thus,
The central angle subtended by the arc equal in length to the radius, i.e. l = r,
we call it radian.
Thus, the angle a = 1 equals in radians,
 
or arc1 = 0.01745329. Arc is abbreviation from Latin arcus, (p = 3.1415926535...).
Protractor - an instrument for measuring angles
Mentioned relations between units of measurement of an angle and arc clearly shows the protractor shown in the below figure marked with radial lines indicating degrees, radians and rarely used gradians (the angle of an entire circle or round angle is 400 gradians).
A right angle equals 100 grad (gradians).
The hundredth part of a right angle is 1g grad, and one 100th part of 1grad is centesimal arc minute 1c, and one 100th part of centesimal arc minute is centesimal arc second 1cc, therefore
Degrees to radians and radians to degrees conversion examples
Example:   Convert 6718 45" to radians.
Solution:  The given angle we write in the expanded notation and calculate its decimal equivalent,
then use the formula to convert degrees to radians
Using a scientific calculator, the given conversion can be performed almost direct.
Before a calculation choose right angular measurement (DEG, RAD, GRAD) by pressing DRG key, then input,    67.1845  INV  DEG   67.3125
Because a calculator must use degrees divided into its decimal part one should press DEG (or  DD) to get decimal degrees. Then press    INV  DRG to get radians,   1.174824753rad
Example:   Convert 2.785rad  to degrees, minutes and seconds.
Solution:  Using formula,
The same result one obtains with a calculator through the procedure, press DRG key to set  RAD measurement, then input   2.785   INV  DRG   177.2986066 grad = 177g29c86cc, press again  INV  DRG   159.5687459 obtained are decimal degrees (DEG), and to convert to degrees/minutes/seconds press  INV   DMS   to get  159 34 7.48".
Example:   Find the length of the arc l that subtends the central angle a = 12338 27" in the circle of radius r = 15 cm.
Solution:  First express the angle a in decimal degrees, i.e.
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