What are trigonometric identities?

• Some are given in the formulae booklet
  • Be sure to note the difference between the?±?and???symbols!

  

How do I know which trig identities to use?

• There is no set method
• This is a matter of experience, familiarity and recognition
  • Practice as many questions as possible
  • Be familiar with trigonometric functions that can be integrated easily
  • Be familiar with common identities – especially?squared?terms
  • sin²?x,?cos²?x,?tan2?x,?cosec²?x,?sec²?x,?tan^2?x?all appear in identities

How do I integrate tan², cot², sec²?and cosec²?

• The?integral?of?sec²x?is?tan?x?(+c)
  • This is because the?derivative?of tan x is sec²x
• The?integral?of?cosec²x?is?-cot?x?(+c)
  • This is because the?derivative?of cot x is -cosec²x
• The?integral?of?tan²x?can be found by using the identity to rewrite tan²x?before integrating:
  • 1 + tan²x?= sec²x
• The?integral?of?cot²x?can be found by using the identity to rewrite cot²x?before integrating:
  • 1 + cot²x?= cosec²x

How do I integrate sin and cos?

• For functions of the form?sin kx,?cos kx?… see Integrating Other Functions
• sin kx × cos kx?can be integrated using the identity for?sin 2A
  • sin 2A = 2sinAcosA

• sinn?kx cos kx?or?sin kx cosn?kx?can be integrated using?reverse chain rule?or?substitution
• Notice no identity is used here but it looks as though there should be!

• sin^2?kx and cos²?kx can be integrated by using the identity for cos 2A
  • For sin²?A, cos 2A = 1 - 2sin²?A
  • For cos²?A, cos 2A = 2cos²?A – 1

How do I integrate tan?

• This is a standard result from the formulae booklet

How do I integrate other trig functions?

• The formulae booklet lists many standard trigonometric derivatives and integrals
  • Check both the “Differentiation” and “Integration” sections
  • For integration using the "Differentiation" formulae, remember that the integral of?f'(x)?is?f(x)?!

• Experience, familiarity and recognition are important – practice, practice, practice!
• Problem-solving techniques