Inverse Circular

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Chapter 35
Inverse Circular Functions
Prepared by Tan Chor How (B.Sc)
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Some fundamental concepts
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Let
y sinx
then we have
or
i.e.
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is the inverse function of
Iff y is the 1-1 function!
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doesnt mean
Also doesnt mean
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1
0
-1
In this region, , y is 1-1
function.
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Now, if you flip the previous graph,
Principal values
-1
The principal values of y is defined as that
value lying between .
0
1
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Similarly, check the cosine graph
1
0
-1
In this region, , y is 1-1
function.
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Now, if you flip the previous graph,
Principal values
-1
The principal values of y is defined as that
value lying between 0 and ? .
0
1
10
0
11
Graph of
Principal values
The principal values of y is defined as that
value lying between .
0
12
Some books write as
. Domain of y is
Range of y is
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In general, we have
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We also have
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e.g. 1
Evaluate .
Soln
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e.g. 2
Evaluate .
Soln
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e.g. 3
Evaluate .
Soln
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e.g. 4
Evaluate .
Soln
Let
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Now, let see Same as
. Domain of y
is Range of y is
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In general, we have
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We also have
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e.g. 5
Evaluate .
Soln
Between
gives .
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Now, let see Same as
. Domain of y
is Range of y is
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Now, let see Same as
. Domain of y
is Range of y is
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e.g. 6
Evaluate .
Soln
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e.g. 7
Evaluate .
Soln
Let
5
4
3
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e.g. 8
Find the value of the following Expression
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Soln
Let and
2
5
3
1
4
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e.g. 9
Find the value of the following Expression
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Soln
Let
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There are 2 possible answers.
because a and b are both positive values, ab
must be positive value.
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Inverse trigonometric identities
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Identity (1)
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Identity (2)
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Let prove the identity 1
To prove
Same as to prove
A
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Check slide 14
LHS of A
RHS of A
We have, and
B
x(-1)
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C
B and C state that both and
are .

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i.e.
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Let prove the identity 2
To prove
Same as to prove
A
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But
and
x(-1)
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Both and
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e.g. 10
Prove that
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Soln
Let
then
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i.e.
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e.g. 11
Prove that
Soln
Let
LHS
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RHS
2
B
3
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Inverse trigonometric equations
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Do keep in mind
Equation Range of solution The only solution




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e.g. 12
Solve the equation .
Soln
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e.g. 13
Solve the equation .
Soln
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e.g. 14
Solve the equation
, assuming that all
the inverse tangents are positive acute angles.

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Soln
Let
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reject
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Differentiation of inverse circular functions
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Differentiation of an inverse function
If
then
The inverse function is
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i.e.
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So, in general,
is the inverse function of
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Differentiation of an inverse circular function
Its inverse function
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So,
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Another way to derive this formula
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Similarly,
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e.g. 15
Find the differentiation of y.
Soln
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e.g. 16
Find dy/dx if .
Soln
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Some important standard types of integral
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(1)
(2)
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In a general format
(1)
(2)
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e.g. 17
Evaluate .
Soln
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Integrals of the form
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This type of integral can always be reduced to
one of the three standard forms
, ,
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e.g. 18
Evaluate .
Soln
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e.g. 19
Evaluate .
Soln
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Integrals of the form
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e.g. 20
Evaluate .
Soln
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e.g. 21
Evaluate .
Soln
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Home works
Mathematics 3 (Further Mathematics) Ex 15a, Ex
15d, Misc Ex.