Edexcel A Level Further Maths: Core Pure:復習筆記5.2.4 Calculus involving Inverse Trig
Differentiating Inverse Trig Functions
What are the inverse trigonometric functions?
- arcsin,?arccos?and?arctan?are functions defined as the inverse functions of sine, cosine and tangent respectively
?which is equivalent to 
?which is equivalent to 
What are the derivatives of the inverse trigonometric functions?
- Unlike other derivatives these look completely unrelated at first
- their derivation involves use of the identity
- hence the squares and square roots!
- their derivation involves use of the identity
- All three are given in the?formula?booklet
- Note with the derivative of
?that 
?is the same as 
How do I show or prove the derivatives of the inverse trigonometric functions?
- For
- Rewrite,
- Differentiate implicitly,
- Rearrange,
- Using the identity
?rewrite, 
- Since,
- Rewrite,
- Similarly, for
- Notice how the derivative of
is positive but is negative for?
- This subtle but crucial difference can be seen in their graphs
- has a positive gradient for all values of
?in its domain - has a negative gradient for all values of ?in its domain
- This subtle but crucial difference can be seen in their graphs
Exam Tip
- For
?the terms on the denominator can be reversed (as they are being added rather than subtracted)
- Don't be fooled by this, it sounds obvious but on awkward "show that" questions it can be off-putting!
Worked Example
a)? ? ? ?Show that the derivative of ?is 
b) Find the derivative of 
.
.
Integrating Inverse Trig Functions
How do I integrate inverse trig functions?
- Use?integration by parts in the same way you would integrate
- These can be integrated using parts however
- rewrite as the product ‘
’ and choose 
?and 
- 1 is easy to integrate and the inverse trig functions have standard derivatives listed in the formula booklet
- rewrite as the product ‘
- The expression
?integrates to 
- The expression
?integrates to 
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