Trigonometric Identities

Here are sixteen Trigonometric Identities:

SIN = Sine
COS = Cosine
TAN = Tangent
CSC = CoSecant
SEC = Secant
COT = CoTangent

#1
             
          1
SIN Θ = -----
        CSC Θ 

#2
          1
COS Θ = -----
        SEC Θ 

#3
        SIN Θ 
TAN Θ = -----
        COS Θ 

#4
        COS Θ     1 
COT Θ = ----- = -----
        SIN Θ   TAN Θ 

#5
          1
SEC Θ = -----
        COS Θ 

#6
           1
CSC Θ  = -----
         SIN Θ 

#7

SIN2 Θ + COS2 Θ = 1

#8

SEC2 Θ = 1 + TAN2 Θ 

#9

CSC2 Θ = 1 + COT2 Θ 

#10

SIN2 Θ = ½ ( 1 - COS 2Θ )

#11

COS2 Θ = ½ ( 1 + COS 2Θ )

#12

SEC2 Θ = 1 + TAN2 Θ

#13

CSC2 Θ = 1 + COT2 Θ

#14

SIN Θ COS Θ = ½ SIN 2Θ

#15

COS 2Θ = COS2 Θ - SIN2 Θ

#16
          2 TAN Θ
TAN 2Θ = ----------
         1 - TAN2 Θ

These trigonometric identities are particularly useful in the electronics field.

The sum formulae are given by

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b).

By taking linear combinations of these, we obtain the product formulae.

2sin(a)cos(b) = sin(a+b) + sin(a-b)
2cos(a)cos(b) = cos(a+b) + cos(a-b)
2sin(a)sin(b) = cos(a-b) - cos(a+b).

These results can be proved using a geometric argument, or by using the exponential forms of cosine and sine, namely

2cos(x) = exp(ix) + exp(-ix)
2isin(x) = exp(ix) - exp(-ix).

They are immensely useful across most of applied mathematics. One of the most simple consequences of them are the double angle formulae, obtained by putting b = a:

sin(2a) = 2sin(a)cos(a)
cos(2a) = cos2(a) - sin2(a).

They can also be used to verify de Moivre's Theorem.

Trigonometric identities are mathematical identities that involve trigonometric functions. Wow... that was deep. What this means is that for any values for each variable you enter into the equation, the equation is either true or undefined.

The half angle identities (ever wanted to find sin(pi/12) ?) are as follows:

              _____________
             /  1 - cos x
sin(x/2)=   /   ---------
          \/        2

              _____________
             /  1 + cos x
cos(x/2)=   /   ---------
          \/        2

              _____________
             /  1 + cos x
tan(x/2)=   /   ---------
          \/    1 - cos x

              _____________
             /      2
csc(x/2)=   /   ---------
          \/    1 - cos x

              _____________
             /      2
sec(x/2)=   /   ---------
          \/    1 + cos x

              _____________
             /  1 - cos x
cot(x/2)=   /   ---------
          \/    1 + cos x

Also, the addition and double angle identities for tangent are:

            tan a + tan b
tan(a+b)= ------------------
          1 - (tan a)(tan b)

          2 tan a
tan(2a)= ---------
         1 - tan2a

Another set of identities worth knowing are the factor formulae. Since sin(a+b) != sin(a) + sin(b), these rules have been derived.

                       a + b     a - b
sin(a) + sin(b) = 2sin ----- cos -----
                         2         2  


                       a + b     a - b
sin(a) - sin(b) = 2cos ----- sin -----
                         2         2  


                       a + b     a - b
cos(a) + cos(b) = 2cos ----- cos -----
                         2         2


                       a + b     a - b
cos(a) - cos(b) = 2sin ----- sin -----
                         2         2

More likely than not, in a practical situation the equation acos(θ) + bsin(θ) will crop up. In such cases, the harmonic form is necessary.

The basic structure of the harmonic form is either Rcos(θ ± α) or Rsin(θ ± α). (R > 0, α is acute). This is best explained with an example:

To express 3cosθ - 4 sinθ in the form Rcos(θ + α):

   Let 3cosθ - 4sinθ ≡ Rcos(θ + α)
                      ≡ R(cosθcosα - sinθsinα)
                      ≡ Rcosθcosα - Rsinθsinα

Now equate the coefficients of cosθ and sinθ to obtain

       3 = Rcosα (1)    and     4 = Rsinα (2)

Squaring (1) and (2) and adding give:

   R2cos2α + R2sin2α = 32 + 42
   R2(cos2α + sin2α) = 25
         ∴ R2 = 25 (since cos2α + sin2α = 1)
          ∴ R = 5 (since R > 0)

Dividing (2) by (1) gives

    Rsinα   4
    ----- = -
    Rcosα   3

           4
  ∴ tanα = -
           3

  ∴ α = 53.1°

Therefore, we have

        3cosθ - 4sinθ = 5cos(θ + 53.1°)

The explanations in this section come from my own study. The specific example above comes from Introducing Pure Mathematics, Smedley/Wiseman.