First Principles Differentiation - Trigonometry (OCR A Level Maths: Pure)

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First Principles Differentiation - Trigonometry

How do I derive the derivatives of trigonometric functions from first principles?

  • Recall that for a function f(x) the definition of the derivative from first principles (see First Principles Differentiation) is:

straight f apostrophe left parenthesis x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis x plus h right parenthesis minus straight f left parenthesis x right parenthesis over denominator h end fraction

  • The derivatives of the trigonometric functions depend on the following small angle approximations
    • When θ is  small (i.e. close to zero) and measured in radians then

sin theta space almost equal to space theta
cos theta space almost equal to space 1 minus 1 half theta squared
tan theta space almost equal to space theta

  • The small angle approximations allow us to produce the following intermediate limit results:

1st Princ Trig Diff Illustr 3_forms, AS & A Level Maths revision notes

1st Princ Trig Diff Illustr 3_derivs, AS & A Level Maths revision notes

  • And those intermediate results allow us to find the derivatives of sin and cos

1st Princ Trig Diff Illustr 4_forms, AS & A Level Maths revision notes

 

7-3-1-1st-princ-trig-diff-illustr-4-derivs

Derivatives of other trigonometric functions

  • The derivative of tan is given by the following formula:

1st Princ Trig Diff Illustr 5, AS & A Level Maths revision notes 

  • The easiest way to derive this is to use the quotient rule and the derivatives of sin and cos
  • But it can also be derived from first principles using the small angle approximation for tan (see the Worked Example)
  • The general formulae for the derivatives of the trigonometric functions are: 1st Princ Trig Diff Illustr 6, AS & A Level Maths revision notes
  • These formulae follow from combining the derivatives of the three basic functions with the chain rule, but they are worth knowing on their own

Examiner Tip

  • Remember that when doing calculus with trigonometric functions you have to measure angles in radians.
  • The formula for the derivative of tan x is included in the exam formulae booklet.
  • The derivatives of sin x and cos x are NOT included in the formula booklet – you have to know them.
  • The small angle approximations for cos x, sin x and tan x are included in the exam formulae booklet – you don't have to memorise them.
  • Be sure to read first principle differentiation exam questions clearly – they will state any results you can treat as 'givens' in your answer. 

Worked example

1st Princ Trig Diff Example, AS & A Level Maths revision notes

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.