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 + 3thus,
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.
Trigonometry contents A