Trigonometry
Basic relationships between trigonometric functions of the same angle
Basic relationships between trigonometric functions of the same angle shown in the unit circle
Basic relationships between trigonometric functions of the same angle, examples
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. Thus,
By plugging (1) and (2) into,
Dividing the basic identity by cos2 x,     sin2 x + cos2 x = 1 | ¸ cos2 x
Dividing the basic identity by sin2 x,     sin2 x + cos2 x = 1 | ¸ sin2 x
Basic relationships between trigonometric functions of the same angle shown in the unit circle
Given 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 of the same angle, 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 nominator and denominator by cos2 x,
 Example:   Prove the identity
 Solution:
Example:   Given sin x + cos x = a,  find  sin4 x + cos4 .
Solution:   Since   sin2 x + cos2 x = 1  then,
(sin2 x + cos2 x)2 = sin4 x + cos4 + 2sin2 x · cos2 x = 1   or   sin4 x + cos4 = 1 - 2sin2 xcos2 x.
As given  sin x + cos x = a  then,   (sin x + cos x)2 = a2   or   sin2 x + cos2 x + 2sin x cos x = a2
therefore,  sin x · cos x = (a2 - 1)/2   and it follows that    sin4 x + cos4 = 1 - (a2 - 1)2/2.
Pre-calculus contents F