Trigonometry
      Trigonometric functions of arcs from 0 to 2p
      Trigonometric functions of negative arcs or angles
      Trigonometric functions of complementary angles
      Trigonometric functions of supplementary angles
Trigonometric functions of arcs from 0 to 2p
Trigonometric functions of negative arcs or angles
We say that arcs x and x are opposite if
x + x = 0  or  x = -x.
Comparing the corresponding sides of the congruent
right-angled triangles in the right figure,
OPxP  and  OPxP
OSxS1  and  OSxS1
OSyS2  and  OSyS2
follows that we can express trigonometric functions of an 
negative arc (-x) by corresponding function of opposite arc 
x, that is
Example:   Given trigonometric functions of negative angle, arc or number should be expressed by the same functions of the positive angle, arc or number.
a)  sin (-200),      b)  cos (-14p/3),      c)  tan (-11),      d)  cot (-750).
Solution:  First express the angle a in decimal degrees, i.e.
a)  sin (-200) = - sin 200
c)  tan (-11) = - tan11 = - tan (3p + 1.575222...) = - tan 1.575222...
d)  cot (-750) = - cot 750 = - cot (2 360 + 30) = - cot 30.
Trigonometric functions of complementary angles
Two angles, x and p/2 - x which form the right angle, are said to be complementary.  
Thus, comparing the corresponding sides of the congruent right-angled triangles in the below figure,
OPxP  and  OPxP,       OSxS1  and  OSxS2    and     OSyS2  and  OSyS1
Example:   The trigonometric functions of the given angle or arc should be expressed by corresponding function of the complementary angle.
a)  sin 30,      b)  cos (p/2 - p/3),      c)  tan 1,      d)  cot 530.
Solution:   a)  sin 30 = cos(90 - 30) = cos 60
 
                  d)  cot 530 = tan (90 - 530) = tan (- 440) = - tan 440 = - tan (2 180 + 80)
                                     = - tan 80 = - cot (90 - 80) = - cot 10.
Example:   Simplify the expression
Solution:   
Trigonometric functions of supplementary angles
Two angles, x and p - x, which when added form a straight
angle, are said to be supplementary.
Comparing the corresponding sides of the congruent right-angled 
triangles in the right figure,
  PxP = PxP   =>   sin (p - x) = sin x  
  OPx = -OPx   =>    cos (p - x) = -cos x  
  SxS1 = -SxS1   =>    tan (p - x) = -tan x  
  SyS2 = -SyS2   =>     cot (p - x) = -cot x  
 
Example:   Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of the supplementary angle.
a)  sin 5p/6,      b)  cos (-320),      c)  tan (p - 1),      d)  cot 30.
Solution:   a)  sin 5p/6 = sin (p - 5p/6) = sin p/6
                 b)  cos (-320) = cos (180 - 500) = - cos 500 = - cos (360 + 140)
                                         = cos (180 - 140) = cos 40
                 c)  tan (p - 1) =  - tan 1
                 d)  cot 30 =  - cot (180 - 30) =  - cot 150.
Example:   Calculate,  sin 3p/2 cos(- p) + tan 5p/4.
Solution:  sin 3p/2 cos(- p) + tan 5p/4 = - 1 (- 1) + tan (p + p/4) = 1 + tan p/4 = 1 + 1 = 2.
Example:   Calculate, 
Solution:
Example:  Prove the identity,
cos2 p/3 sin (p/2 - x) - cos (p - x) cos2 p/6 = tan (p/2 + x) sin (2p - x).
Solution:  Since  sin (p/2 - x) = cos x,   cos (p - x) = - cos x,   tan (p/2 + x) = - cot x
and  sin (2p - x) = - sin x  then,      
(cos p/3)2 cos x - (- cos x) (cos p/6)2 = - cot x (- sin x),
(1/2)2 cos x + (3/2)2 cos x = (cos x/sin x) sin x  =>  cos x = cos x.
Example:  Prove the identity, 
cot2 (p + x) cos2 (p/2 + x) + sin (- x) sin (p + x) = tan (2p - x) cot (- x).
Solution:    [cot (p + x)]2 [cos (p/2 + x)]2 + (- sin x) sin (p + x) = (- tan x) (- cot x),
(cot x)2 (- sin x)2 + (- sin x) (- sin x) = (sin x/cos x) (cos x/sin x)
cos2 x + sin2 x = (sin x/cos x) (cos x/sin x) = 1.
Pre-calculus contents C
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