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 shown in the tabular form
         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 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 shown in the tabular form
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 the numerator and the denominator by cos2 x,
Example:   Prove the identity
Solution:  
Example:   Given sin x + cos x = a,  find  sin4 x + cos4 x .
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.
Trigonometry contents A
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