Formulary Trigonometry




Basic formulas

 
sec(t) = 1/cos(t)

csc(t) = cosec(t) = 1/sin(t)

cos2(t) + sin2(t) = 1

1 + tan2(t) = sec2(t)

1 + cot2(t) = csc2(t)

Periodicity

2.pi for cos(t) and sin(t)

pi for tan(t) and cot(t)

Special values

 
sin(pi/3) = sqrt(3)/2
cos(pi/3) =  1/2

cos (pi/4) =  sin(pi/4) = sqrt(1/2)

cos (pi/6) = sqrt(3)/2
sin(pi/6) =  1/2.

Related values

t and t' are supplementary values <=> t+t' = pi.
If t and t' are supplementary values then
 
       sin(t) = sin(t')
cos(t) = -cos(t')
tan(t) = -tan(t')
cot(t) = -cot(t')
t and t' are complementary values <=> t+t' = pi/2.
If t and t' are complementary values then
 
       sin(t) = cos(t')
cos(t) = sin(t')
tan(t) = cot(t')
cot(t) = tan(t')
t and t' are opposite values <=> t+t' = 0.
If t and t' are opposite values then
 
       sin(t) = -sin(t')
cos(t) = cos(t')
tan(t) = -tan(t')
cot(t) = -cot(t')
t and t' are anti-supplementary values <=> t-t' = pi.
If t and t' are anti-supplementary values then
 
       sin(t) = -sin(t')
cos(t) = -cos(t')
tan(t) = tan(t')
cot(t) = cot(t')

Triangle ABC

Angle A is the right angle of the triangle ABC, then
 
        sin(B) = b/a   cos(B) = c/a  tan(B) = b/c

        cos(C) = b/a   sin(C) = c/a   tan(C) = c/b
In any triangle:
Sine rule
 
          a         b        c
        ------ =  ------ = ------
        sin(A)    sin(B)   sin(C)
Cosine rule
 
        a2  = b2  + c2  - 2 b c cos(A)

        b2  = c2  + a2  - 2 c a cos(B)

        c2  = a2  + b2  - 2 a b cos(C)
Area = (1/2).a.c.sin(B)= (1/2).b.c.sin(A) = (1/2).a.b.sin(C)

Sum formulas

 
cos(u - v) = cos(u).cos(v)+sin(u).sin(v)

cos(u + v)  =  cos(u).cos(v)-sin(u).sin(v)

sin(u - v) = sin(u).cos(v)-cos(u).sin(v)

sin(u + v) = sin(u).cos(v)+cos(u).sin(v)

           tan(u) + tan(v)
tan(u+v) = -----------------
           1 - tan(u).tan(v)

           tan(u) - tan(v)
tan(u-v) = -----------------
           1 + tan(u).tan(v)

Doubling Formulas

 
sin(2u) = 2sin(u).cos(u)

cos(2u) = cos2 (u) - sin2 (u)


            2 tan(u)
tan(2u)  =  -----------
            1- tan2(u)

Carnot formulas

 
1 + cos(2u) = 2 cos2 (u)

1 - cos(2u) = 2 sin2 (u)

t-formulas

 
Let  t = tan(u) , then


           1 - t2
cos(2u) = ---------  ;
           1 + t2

             2t
sin(2u) =  -------- ;
            1 + t2

              2t
tan(2u) =  -------  ;
            1 - t2

Simpson formulas

 
cos(x) + cos(y) = 2 cos((1/2)(x + y)) cos((1/2)(x - y))

cos(x) - cos(y) = -2 sin((1/2)(x + y)) sin((1/2)(x - y))

sin(x) + sin(y) = 2 sin((1/2)(x + y)) cos((1/2)(x - y))

sin(x) - sin(y) = 2 cos((1/2)(x + y)) sin((1/2)(x - y))

Basic equations

 
  cos(u) = cos(v)
<=>
  (u = v + k.2pi) of (u = -v + k.2pi)


  sin(u) = sin(v)
<=>
  (u = v + 2.k.pi) of (u = pi - v + 2.k.pi)


   tan(u) = tan(v)
<=>
  (u = v + k.pi)  on condition that  tan(u) and tan(v) exist.


  cot(u) = cot(v)
<=>
  (u = v + k.pi)  on condition that  cot(u) and cot(v) exist.




Topics and Problems

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