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Trigonometry
 Trigonometric functions of arcs from 0 to ± 2p
 Values of trigonometric functions of arcs p/6, p/4 and p/3
 The given arcs are, one twelfth, one eighth and one sixth of the circumference 2p of the unit circle so the coordinates of terminal points of the arcs are the elements of the equilateral triangle with the side a = 1 (Fig.a, c) and the sides of the square with diagonal d = 1, (Fig. b).
 The values of the trigonometric functions of arcs that are multipliers of 30° (p/6) and 45° (p/4)
 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 acute angle a between the x-axis and the terminal side of angle x we call the reference angle.
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
These 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.
 Basic relationships between trigonometric functions of the same angle

Using the Pythagorean theorem for the right triangle in the figure we get the fundamental trigonometric identity

sin2 x + cos2 x = 1

which, because of periodicity of the trigonometric functions, holds for an arbitrary angle x = a + k · 2p  therefore,

sin2 x + cos2 x = 1,   x Î R.

From this identity and the definitions of the functions, tangent

and cotangent
we can derive twelve formulas so that each function is expressed through another three.
 Basic relationships between trigonometric functions of the same angle shown in the unit circle

Relations between the trigonometric functions of the same angle, expressed by absolute value, are included in the definitions of the functions in right triangles shown in Fig. a - d, and included in the table below.

 Basic relationships between trigonometric functions shown in the tabular form
 Trigonometric expressions examples

Example:  Find values of other trigonometric functions of an angle a, if given  sin a = - 4/5  and          270° < a < 360°.

Solution:  Since a is a forth-quadrant angle, then

Example:  Find the value of

Solution:  Dividing the numerator and the denominator by cos2 x,

 
 
 
 
 
 
 
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