Mnemonic Device for Differentiation and Integration of Trigonometric functions.

Question originally asked by Chantey Peter.

First lets us look at the differentiation of Trigonometric functions:

Since, I have applied learning/remembering by pattern in this Mnemonic Device. From this I have developed simple rules.

Rules derived from pattern.

1. If there is C in front : then negative (-)  on the backside.

Eg. Derivative of cosx = - sin x ; Since there is C in Cosx so negative on the backside.

2. If there is C in the last letter of the front word : then there are two items.

  Eg. Derivative of secx = secx.tanx ; Since there is C at last in seCx so there are two items as   secx and tanx.

3. If there are 2 terms ; one must be itself.

Now, let’s talk the individuals:

a.      Derivative of Sinx = cosx  ; you should and would remember this automatically.

If not: remember that there is a formula where sin^2x + cos^2x = 1. So sin and cos comes together.

b.      Derrivative of Cosx = - Sinx ; Sin and cos go on exchange according to a. And Negative sign according to rule number 1.

c.       Derrivative of tanx = sec^2x ; Do you remember the formula sec^2x + tan^2x = 1. From this you can remember that sec and tanx comes together.

d.      Derivative of cosecx = - cosecx.cotx ; negative is according to rule 1. And, there are 2 terms according to rule 2. Now, Remember there are 2 terms, cosecx and cotx : cosecx is OK since it is in question ( analyse the pattern ) and Cotx from the last.

e.      Derivative of secx = secx.tanx ; two terms according to rule 2. Secx : since it is in the question, tanx: last from sin cos and tan. Just opposite of cosecx. ( analyse the pattern )

f.        Derivative of cotx = -cosec^2x  ; Negative is according to rule number 1. And since tanx had sec^2x ; cotx must have cosec^2x. ( anaylze the pattern )

Now let’s remember Integration of Trigonometric Functions.

Basic fact to remember --- integration is opposite of differentiation/derivative

Rules derived from pattern 

       1.      If C in second part then negative on the result. ( opposite of derivative )

       2.      If there is C at last then there are two terms in result. ( same as in derivative )

       3.      If there are two terms one must be itself.
  

     Now, let’s talk the individuals:

        a.      Integration of sinx  = - cosx + c ; Negative for rule 1. remember the relation of sin and cos.

        b.      Integration of cosx = sinx + c ; Relation of sin and cos.

        c.       Integration of tanx = -ln (cosx) + c ; If you know the rule of integration then you will

        d.      Integration of cosecx = - ln ( cosecx + cotx ) + c ; Two terms according to rule 2. Negative according to rule 1. Since there are two terms  one is cosec and other is of last cot. Same as in Derivative. Read the pattern.

        e.      Integration of secx = ln ( secx + tanx ) + c ; Two terms according to rule number 2. Since there are two terms one must be itself ( rule no. 3 ). Other is last one i.e. tanx. Read the pattern.

        f.        Integration of cotx = ln ( sinx ) + c ; Since tan had cos; cot has sin. Read the pattern.

     
      Since it is based on reading the pattern, you may get confused by reading this. But use there on some places then you will surely remember the pattern. Also, the best way to remember maths formula is to practice !!!