Trigonometry
Trigonometric Functions Unit of measurement of angles - a radian (the circular measure)
Protractor - an instrument for measuring angles 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). 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 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 · 2p,  k = 0, ±1, ±2, ±3, . . . or   x = a° + k · 360°,   k Î 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.   Pre-calculus contents C 