Trigonometry
     Trigonometric Functions
      The unit circle or the trigonometric circle
         Division of the circumference of the unit circle to the characteristic angles
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.
Pre-calculus contents C
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