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
      The unit circle or the trigonometric circle
         Division of the circumference of the unit circle to the characteristic angles
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 67 18 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 = 123 38 27" in the circle of radius r = 15 cm.
Solution:  First express the angle a in decimal degrees, i.e.
The unit circle or trigonometric circle
A circle of radius r = 1, with the center at the origin O(0, 0) of a coordinate system, we call the unit or trigonometric circle, see the figure below.
The arc of the unit circle that describes a point traveling anticlockwise (by convention, clockwise is taken to be negative direction) from the initial position P1(1, 0) on the x-axis, along the circumference, to the terminal position P equals the angular measure/distance x = arad, in radians.
An angle is in standard position if its initial side lies along the
positive x-axis.
If we take the positive direction of the x-axis as the beginning 
of a measurement of an angle (i.e., a = 0rad, both sides of
an angle lie on the x-axis), and the unit point P1 as the initial
point of measuring the arc, then the terminal side of an angle,
which passes through the terminal point P of the arc, 
rotating around the origin (in any direction) describes different 
angles, and the terminal point P corresponding arcs,
x = arad + k 2pk = 0, 1, 2, 3, . . . .
 or   x = a + k 360,   k Z.
It means that every arc x ends in the same point P in which ends the corresponding arc a.
Thus, at any point P on the circumference of the unit circle end infinite arcs x = a + k 2p, which differ by the multiplier 2p, and any number x associates only one point P.
Division of the circumference of the unit circle to the characteristic angles
There is a common division of the circumference of the unit circle to the characteristic angles or the corresponding arcs which are the multipliers of the angles, 30 (p/ 6) and 45 (p/ 4).
We can say that a unit circle is at the same time  
numerical circle.
The numerical circle shown in the right figure is formed
by winding the positive part of number line, with the unit
that equals the radius, around the unit circle in the 
anticlockwise direction and its negative part in clockwise
direction.
Therefore, the terms angle, arc and number in the
trigonometric definitions and expressions are mutually
interchangeable.
 
Example:   In which quadrant lies second or the terminal side of the angle x = 1280
Solution:  Dividing the given angle by 360 we calculate the
number of rotations, or round angles, described by terminal side
of the angle x, and the remaining angle a position of which we
want to find.
 
since  x = a + k 360  then   k = 3  and  a = 200.
therefore, terminal side of the angle x lies in the third quadrant.
Example:   In which quadrant lies the endpoint of the arc x = - 47p/3 of a unit circle.
Solution:  Given arc
can be expanded to
 
Thus, the endpoint of the arc x moved around a unit circle in the clockwise (negative) direction 7 times and described additional arc a = - (5/3)prad, so its endpoint P lies in the first quadrant.
Trigonometry contents A
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