TRIGONOMETRY

Basic formulas

cos2(t) + sin2(t) = 1

1 + tan2(t) = sec2(t)

1 + cot2(t) = cosec2(t)

Area of a triangle


Area of a triangle
 
            

The area of the triangle is a.h/2 .
But in triangle BAH, we have sin(B) = h/c .
Hence the area of the triangle is a.c.sin(B) /2.
Similarly, we have that the area of the triangle 
= b.c.sin(A) /2 = a.b.sin(C) /2


The area of a triangle ABC =(1/2) a.c.sin(B) = (1/2) b.c.sin(A) = (1/2) a.b.sin(C)




You can also use Heron's formula to calculate the area of a triangle.

 
Let s = half the circumference of the triangle = (a +b + c)/2.

The area of a triangle ABC =
     ______________________________
    V s (s - a) (s - b) (s - c)








Sine rule


          a         b        c
        ------ =  ------ = ------
        sin(A)    sin(B)   sin(C)

Let R be the radius of the circle with center O through the points A,B and C.

In any triangle ABC we have
 
          a         b        c
        ------ =  ------ = ------ = 2R
        sin(A)    sin(B)   sin(C)

Cosine rule


In any triangle ABC we have

 
        a2  = b2  + c2  - 2 b c cos(A)

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

        c2  = a2  + b2  - 2 a b cos(C)






Sum formulas


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



t-Formulas or Half-Angle Formulas


Let t = tan(u) , then


           1 - t2
cos(2u) = ---------
           1 + t2
or
             1 -  tan2(u)
cos(2u) =   -------------
             1 + tan2 (u)

             2t
sin(2u) =  --------
            1 + t2
or
             2 tan(u)
sin(2u) =  -----------
           1 + tan2 (u)




Simpson formulas

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

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

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

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