Trigonometry
     Trigonometric Functions
      Definitions of trigonometric functions
      Periodicity of trigonometric functions
      Signs of trigonometric functions
         The table of signs of trigonometric functions
Definitions of trigonometric functions
Let x be an arc of the unit circle measured counterclockwise from the x-axis. It is at the same time the circular measure of the subtended central angle a as is shown in the below figure.
In accordance with the definitions of trigonometric functions
in a right-angled triangle,
- the sine of an angle a (sina) in a right triangle is the ratio  of the side opposite the angle to the hypotenuse.
- the cosine of an angle a (cosa) in a right triangle is the ratio of the side adjacent to it to the hypotenuse.
Thus, from the right triangle OPP, follows
sinx = PP    The sine of arc x is the ordinate of the arc 
                    endpoint.
cosx = OP  The cosine of arc x is the abscissa of the arc 
                    endpoint.
The tangent of an angle a (tana) in a right triangle is the ratio of the lengths of the opposite to the adjacent side.
The cotangent is defined as reciprocal of the tangent, thus
From the similarity of the triangles OPP and OP1S1,
Hence, the definition of the tangent function in the unit circle,
tanx = P1S1    The tangent of an arc x is the ordinate of intersection of the second or terminal side (or its extension) of the given angle and the tangent line x = 1.
From the similarity of the triangles, OPP and OP2S2,
cotx = P2S2    The cotangent of an arc x is the abscissa of intersection of the second or terminal side (or its extension) of the given angle and the tangent y = 1.
It is obvious from the definitions that the tangent function is not defined for arguments x for which cos x = 0,
as well as the cotangent function is not defined for the arguments for which
sin x = 0.
Periodicity of trigonometric functions
After the argument (arc) x passes through all real values from the interval 0 < x < 2p or after the terminal side of an angle turned round the origin for an entire circle, trigonometric or circular functions will repeat their initial values.
As the terminal point P of an arc continue rotation around a unit circle in the positive direction passing over the initial point P1 , it takes next values from the interval 2p < x < 4p, then the values from the interval 4p < x < 6p and so on.
On the same way we can examine the rotation of the terminal point P of an arc x in the negative (clockwise) direction, when it will pass through the values from the intervals, 0 to -2p, from -2p to -4p, and so on.
It follows that the argument x can take any value,
x = arad + k 2pk = 0, +1, +2, +3, . . .   or     x = a + k 360,   k Z.
that is, every real value between - oo and + oo.
Particularly, for k = 0, i.e., during the first rotation the value of argument is  x = arad.
While the arc endpoint continues rounding over the starting point the trigonometric functions will, in every interval of length 2p (i.e., from 2p to 4p, from 4p to 6p, . . . , or from 0 to  - 2p, from  - 2p to -4p, . . .) take the same values in the same order they took in the first interval [0, 2p].
Functions which have the characteristic to take the same values while their argument changes for all integral multiples of a constant interval (or a constant increases in amount called increment) we call periodic functions, and this constant interval we call period.
Hence, we say that trigonometric functions are periodic functions of x, so that
                            f (x) = sin x    and    f (x) = cos x   of the period  P = 2p,
while functions,      f (x) = tan x    and    f (x) = cot x   of the period  P = p.
The periodicity of trigonometric functions show the identities,
  sin (a + k 2p) = sin a    and    cos (a + k 2p) = cos a,   k Z  
  tan (a + k p) = tan a    and    cot (a + k p) = cot a,   k Z  
Signs of trigonometric functions
Quadrant I  - Values of trigonometric functions, sine, cosine, tangent and cotangent of any arc from the first quadrant are all positive as positive are the coordinates of the points, P, S1 and S2 that define their values.
Quadrant II  - For arcs from the second quadrant points, P and S2 both have negative abscissas (see the above figure), so the cosine and cotangent are negative. The ordinate of the terminal point P is positive so that the sine is positive while the ordinate of the point S1 is negative, thus the tangent is negative.
Quadrant III  - As the abscissas and the ordinates of the terminal points P of arcs from the third quadrant (see the above figure) are negative it follows that cosine and sine functions of these arcs are negative. The ordinates of the points S1 and the abscissas of the points S2 that belong to the arcs from the third quadrant are positive. Thus, the tangent and cotangent of these arcs are positive.
Quadrant IV  - The functions, sine, tangent and cotangent of the arcs from the fourth quadrant are negative as are the coordinates of the points, P, S1 and S2, that belong to them. Only the cosine function of arcs from the fourth quadrant is positive as are the abscissas of points P that belong to them (see the above figure).
The table of signs of trigonometric functions
Example:   To which quadrant belongs the endpoint of an arc a if sin a < 0 and cot a > 0.
Solution:  The right figure shows that the ordinate of
the endpoint P of an arc from the third quadrant is 
negative, so sin a < 0 while the abscissa of the 
point (in which the extension of the terminal side of 
the angle a intersects the tangent y = 1) is positive, 
i.e., cot a > 0.
That is in accordance with the signs in the above 
table. 
Trigonometry contents A
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