Calculus involving Inverse Trig | Edexcel A Level Further Maths: Core Pure Revision Notes 2017

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

$\arcsin (\frac{\sqrt{3}}{2}) = \frac{π}{3}$ which is equivalent to $\sin (\frac{π}{3}) = \frac{\sqrt{3}}{2}$
$\arctan (- 1) = \frac{3 π}{4}$ which is equivalent to $\tan (\frac{3 π}{4}) = - 1$

What are the derivatives of the inverse trigonometric functions?

$f (x) = \arcsin x$
- $f' (x) = \frac{1}{\sqrt{1 - x^{2}}}$
$f (x) = \arccos x$
- $f' (x) = - \frac{1}{\sqrt{1 - x^{2}}}$
$f (x) = \arctan x$
- $f' (x) = \frac{1}{1 + x^{2}}$
Unlike other derivatives these look completely unrelated at first
- their derivation involves use of the identity $\cos^{2} x + \sin^{2} x \equiv 1$
- hence the squares and square roots!
All three are given in the formula booklet
Note with the derivative of $\arctan x$ that $(1 + x^{2})$ is the same as $(x^{2} + 1)$

How do I show or prove the derivatives of the inverse trigonometric functions?

For $y = \arcsin x$
- Rewrite, $\sin y = x$
- Differentiate implicitly, $\cos y \frac{d y}{d x} = 1$
- Rearrange, $\frac{d y}{d x} = \frac{1}{\cos y}$
- Using the identity $\cos^{2} y \equiv 1 - \sin^{2} y$ rewrite, $\frac{d y}{d x} = \frac{1}{\sqrt{1 - \sin^{2} y}}$
- Since, $\sin y = x$ , $\frac{d y}{d x} = \frac{1}{\sqrt{1 - x^{2}}}$
Similarly, for $y = \arccos x$
- $\cos y = x$
- $- \sin y \frac{d y}{d x} = 1$
- $\frac{d y}{d x} = - \frac{1}{\sin y}$
- $\frac{d y}{d x} = - \frac{1}{\sqrt{1 - \cos^{2} y}}$
- $\frac{d y}{d x} = - \frac{1}{\sqrt{1 - x^{2}}}$
Notice how the derivative of $y = \arcsin x$ is positive but is negative for $y = \arccos x$
- This subtle but crucial difference can be seen in their graphs
  - $y = \arcsin x$ has a positive gradient for all values of $x$ in its domain
  - $y = \arccos x$ has a negative gradient for all values of $x$ in its domain

Examiner Tip

For $f (x) = \arctan x$ the terms on the denominator can be reversed (as they are being added rather than subtracted)
- $f' (x) = \frac{1}{1 + x^{2}} = \frac{1}{x^{2} + 1}$
- 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 $\arctan x$ is $\frac{1}{1 + x^{2}}$

5-8-3-ib-hl-aa-only-we2a-soltn

Find the derivative of

\arctan (5 x^{3} - 2 x)

5-8-3-ib-hl-aa-only-we2b-soltn

Integrating Inverse Trig Functions

How do I integrate inverse trig functions?

Use integration by parts in the same way you would integrate $\ln x$
These can be integrated using parts however
- rewrite as the product ‘ $1 \times f (x)$ ’ and choose $u = f (x)$ and $\frac{d v}{d x} = 1$
- 1 is easy to integrate and the inverse trig functions have standard derivatives listed in the formula booklet
The expression $\frac{x}{1 + x^{2}}$ integrates to $\frac{1}{2} \ln (1 + x^{2})$
The expression $\pm \frac{x}{\sqrt{1 - x^{2}}}$ integrates to $\mp \sqrt{1 - x^{2}}$

$\begin{array}{rcl} \int \arcsin (x) d x & = & x \arcsin (x) + \sqrt{1 - x^{2}} + c \\ \int arc \cos (x) d x & = & x arc \cos (x) - \sqrt{1 - x^{2}} + c \\ \int arc \tan (x) d x & = & x arc \tan (x) - \frac{1}{2} \ln (1 + x^{2}) + c \end{array}$

Calculus involving Inverse Trig (Edexcel A Level Further Maths: Core Pure)

Revision Note

What are the inverse trigonometric functions?

What are the derivatives of the inverse trigonometric functions?

How do I show or prove the derivatives of the inverse trigonometric functions?

How do I integrate inverse trig functions?

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1. Complex Numbers

1.1 Complex Numbers & Argand Diagrams

1.1.1 Introduction to Complex Numbers

1.1.2 Solving Equations with Complex Roots

1.1.3 Modulus & Argument

1.1.4 Modulus-Argument Form

1.1.5 Loci in Argand Diagrams

1.1.6 Regions in Argand Diagrams

1.2 Exponential Form & de Moivre's Theorem

1.2.1 Exponential Form

1.2.2 de Moivre's Theorem

1.2.3 Applications of de Moivre's Theorem

1.2.4 Roots of Complex Numbers

2. Matrices

2.1 Properties of Matrices

2.1.1 Introduction to Matrices

2.1.2 Determinants of Matrices

2.1.3 Inverses of Matrices

2.2 Transformations using Matrices

2.2.1 Transformations using a Matrix

2.2.2 Geometric Transformations with Matrices

2.2.3 Invariant Points & Lines

3. Further Algebra & Functions

3.1 Roots of Polynomials

3.1.1 Roots of Polynomials

3.1.2 Linear Transformations of Roots

3.2 Series

3.2.1 Sums of Integers, Squares & Cubes

3.2.2 Method of Differences

3.3 Maclaurin Series

3.3.1 Maclaurin Series

4. Hyperbolic Functions

4.1 Hyperbolic Functions

4.1.1 Hyperbolic Functions & Graphs

4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

4.1.3 Hyperbolic Identities & Equations

4.1.4 Differentiating & Integrating Hyperbolic Functions

5. Further Calculus

5.1 Volumes of Revolution

5.1.1 Volumes of Revolution

5.1.2 Modelling with Volumes of Revolution

5.2 Methods in Calculus

5.2.1 Improper Integrals

5.2.2 Mean Value of a Function

5.2.3 Integrating with Partial Fractions

5.2.4 Calculus involving Inverse Trig

5.2.5 Integration by Substitution

6. Further Vectors

6.1 Vector Lines

6.1.1 Equations of Lines in 3D

6.1.2 Pairs of Lines in 3D

6.1.3 Angle between Lines

6.1.4 Shortest Distances - Lines

6.2 Vector Planes

6.2.1 Equations of planes

6.2.2 Combinations of Lines & Planes

6.2.3 Combinations of Planes

6.2.4 Shortest Distances - Planes

7. Polar Coordinates

7.1 Polar Coordinates

7.1.1 Polar Coordinates

7.1.2 Calculus with Polar Coordinates

8. Differential Equations

8.1 First Order Differential Equations

8.1.1 Intro to Differential Equations

8.1.2 Solving First Order Differential Equations

8.1.3 Modelling using First Order Differential Equations

8.2 Second Order Differential Equations

8.2.1 Solving Second Order Differential Equations

8.2.2 Coupled First Order Linear Equations

8.3 Simple Harmonic Motion