Trigonometry
      Trigonometric functions of arcs from 0 to 2p
         Trigonometric functions of arcs that differ on p/2
         Trigonometric functions of arcs that differ on p
         Trigonometric functions of arcs whose sum is 2p
      Trigonometric functions values and identities examples
Trigonometric functions of arcs that differ on p/2
Comparing the corresponding sides of the congruent right-angled 
triangles, in the right figure, we get the relations of trigonometric 
functions of an arc x and the arc p/2 + x 
  PxP = OPx   => sin (p/2 + x) = cos x  
  OPx = -PxP   =>   cos (p/2 + x) = -sin x  
  SxS1 = -SyS2   =>   tan (p/2 + x) = -cot x  
  SyS2 = -SxS1  =>   cot (p/2 + x) = -tan x  
Example:   Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of angle which differ from the given for 90 (p/2).
a)  sin 1,      b)  cos 150,      c)  tan (-7p/4),      d)  cot 50.
Solution:   a)  sin 1 = - cos (p/2 + 1) =  - cos 2.570796...
                 b)  cos 150 = cos (90 + 60) = - sin 60
                 c)  tan (-7p/4) =  - cot (p/2 - 7p/4) =  - cot ( - 5p/4) = cot (p + p/4) = cot p/4
                 d)  cot 50 =  - tan (90 + 50) = - tan 140.
Trigonometric functions of arcs that differ on p
Comparing the corresponding sides of the congruent right-angled 
triangles, in the right figure, we get the relations of trigonometric 
functions of an arc x and the arc p + x 
  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 angle which differ from the given for 180 (p).
a)  sin 235,      b)  cos p/6,      c)  tan (-300),      d)  cot 4.
Solution:   a)  sin 235 = sin (180 + 55) = - sin 55
                 b)  cos p/6 = - cos (p + p/6) = - cos 7p/6
                 c)  tan (-300) =  tan (180 - 300) =  tan (-120) = - tan 120
                 d)  cot 4 = cot (p + 0.858407...) = cot 0.858407....
Example:   Simplify expression  cot (p - x) cos (p/2 + x) + tan (p/2 - x) tan (p + x) - cos (- x)
Solution:   cot (p - x) cos (p/2 + x) + tan (p/2 - x) tan (p + x) - cos (- x) = 
                                                = - cot x ( - sin x) + cot x tan x - cos x = cos x + 1 - cos x = 1.
Trigonometric functions of arcs whose sum is 2p
The right figure shows relations between sides of the congruent 
right-angled triangles as follows, 
  PxP-PxP   =>   sin (2p - x) = -sin x  
OPx = cos (2p - x) = cos x  cos (2p - x) = cos x  
  SxS1 = -SxS1   =>   tan (2p - x) = -tan x  
  SyS2 = -SyS2   =>   cot (2p - x) = -cot x  
Trigonometric functions values and trigonometric identities examples
Example:   Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of angle which when added with a given make 360 (2p).
a)  sin p/3,      b)  cos 1,      c)  tan 330,      d)  cot 10p/11.
Solution:   a)  sin p/3 = -sin (2p - p/3) = - sin 5p/3
                  b)  cos 1 = cos (2p - 1) = cos 5.283185...
                  c)  tan 330 =  tan (360 - 330) = - tan 30
                 d)  cot 10p/11 = - cot (2p - 10p/11) =  - cot p/11.
Example:   Prove that  sin 320 + cos 50 = 0.
Solution:   Since      sin 320 = sin (360 - 40) = - sin 40,    and as    cos 50 = sin 40
                    then      - sin 40 + sin 40 = 0.
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.
Trigonometry contents A
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