Trigonometry
      Calculation of values of trigonometric functions
         Trigonometric reduction formulas - the reference angles
         Calculation of values of trigonometric functions of an arbitrary angle x, examples
Calculation of values of trigonometric functions
Trigonometric reduction formulas, the reference angles
Values of trigonometric functions, of any angle greater than 90 (p/2) can be expressed by the corresponding value of the function of an angle from the first quadrant.
The figures, a, b and c below, show these relations for the angles x whose terminal side falls in the second, third or fourth quadrant.
The acute angle a between the x-axis and the terminal side of angle x we call the reference angle.
Angles x whose terminal side falls in the second quadrant
we denote as,  x = p - a.
  sin (p - a) = sin a  
  cos (p - a) = -cos a  
  tan (p - a) = -tan a  
  cot (p - a) = -cot a  
Angles x whose terminal side falls in the third quadrant we denote as,  x = p + a
Angles x whose terminal side falls in the third quadrant
we denote as,  x = p + a.
  sin (p + a) = sin a  
  cos (p + a) = -cos a  
  tan (p + a) = tan a  
  cot (p + a) = cot a  
Angles x whose terminal side falls in the fourth quadrant
we denote as,  x = 2p - a.
  sin (2p - a) = -sin a  
  cos (2p - a) = cos a  
  tan (2p - a) = -tan a  
  cot (2p - a) = -cot a  
If a given arc x is greater than 2p then, before calculating values of functions sine and cosine, we should divide it by 2p, while for functions tangent and cotangent by p, to determine integral multiplier k and reminder a therefore,
and for functions, tangent and cotangent
Given properties of trigonometric functions are included in the reduction formulas that give the value of any angle x greater than 90 (p/2) in terms of same function of an acute angle a.
We calculate values of trigonometric functions of an arbitrary angle x by using its reference angle a.
If x is an angle in standard position, then its reference angle is given by the acute angle x which is enclosed between the terminal side of the x and the x-axis.
Calculation of values of trigonometric functions of an arbitrary angle x examples
Example:   Calculate  sin 1110.
Solution:   To use the expression x = a + k 360 we should divide given angle by 360 to get an integer multiple k and remaining angle a, thus
Example:   Calculate  cos (- 77p/4).
Solution:   Since cos (- x) = cos x  we can write  cos (- 77p/4) = cos 77p/4
then, divide given x by 2p,    x = 77p/4 = 19p + p/4 = 9 2p + 5p/4  
so that,    cos (- 77p/4) = cos 77p/4 = cos (9 2p + 5p/4) = cos 5p/4.
As the terminal side of the angle 5p/4 lies in the third quadrant we use   cos (p + a) = - cos a
therefore,     cos 5p/4 = cos (p + p/4) = - cos p/4 = - 2/2.
Example:   Calculate  tan 817 35 42.
Solution:   According to  x = a + k 180 we divide the given angle x by 180 to determine k and a, so
x = 817 35 42 = 97 35 42 + 4 180
thus,  tan 817 35 42 = tan (97 35 42 + 4 180) = tan 97 35 42
        as it is second-quadrant angle we use  tan (180 - a) = - tan a
        that is,  180 - a = 179 59 60 - 97 35 42 = 82 24 18 
therefore,    tan 97 35 42 = tan (180 - 82 24 18) = - tan 82 24 18 = - 7.49964. 
When we use a scientific calculator to find values for trigonometric functions, a given angle should be entered as decimal degree in DEG mode, so input
82.2418   INV   DEG    =>    82.405   TAN     =>    7.49964367.
Example:   Calculate  cot 27p/7.
Solution:   Decompose the angle to   x = a + k p, 
                                          that is,    x = 27p/7 = (3 + 6/7)p = 6p/7 + 3p
thus,   cot 27p/7 = cot (6p/7 + 3p) = cot 6p/7
since for a fourth-quadrant angle holds  cot (2p - a) = - cot a
then,     cot 6p/7 = cot (2p - p/7) = - cot p/7 = - 2.07652.
When we use calculator we set it in the RAD mode using DRG key, then enter,
INV  p     7  =  0.44879895    TAN   (0.481574618)     1 / x     =>    2.076521397.   
That is, the value of the function cotangent we calculate as the reciprocal value of the function tangent, as    cot x = 1/tan x.
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