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 = - Ö2/2.
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.   Functions contents D 