Transcript The inverse trigonometric functions

Contents
The inverse trigonometric
functions
The inverse trigonometric functions
The reciprocal trigonometric functions
Trigonometric identities
Examination-style question
1 of 35
© Boardworks Ltd 2006
The
inverse
of
the
sine
function
Suppose we wish to find θ such that
sin θ = x
In other words, we want to find the angle whose sine is x.
This is written as
θ = sin–1 x or θ = arcsin x
In this context, sin–1 x means the inverse of sin x.
This is not the same as (sin x)–1 which is the reciprocal of
sin x, 1 .
sin x
Is y = sin–1 x a function?
The
inverse
of
the
sine
function
y
We can see from the graph
y = sin x
of y = sin x between x = –2π
and x = 2π that it is a manyto-one function:
x
y
The inverse of this graph is
not a function because it is
one-to-many:
y = sin–1 x
x
The
inverse
of
the
sine
function
However, remember that if we use a calculator to find sin x
–1
(or arcsin x) the calculator will give a value between –90° and
90° (or between – 2 ≤ x ≤ 2 if working in radians).
There is only one value of sin–1 x in this range, called the
principal value.
So, if we restrict the domain of f(x) = sin x to –  ≤ x ≤  we
2
2
have a one-to-one function:
y
y = sin x
1



2
2
–1
x
–1 x
The
graph
of
y
=
sin
Therefore the inverse of f(x) = sin x, –  ≤ x ≤  , is also a
2
one-to-one function:
f –1(x) = sin–1 x
sin–1 x
2
y

y = sin–1 x
The graph of y =
2
1
is the reflection of
y = sin x
y = sin x in the line y = x:
x
(Remember the scale
1 2
 2 –1
–1
used on the x- and y-axes
 2
must be the same.)
The domain of sin–1 x is the same as the range of sin x :
–1 ≤ x ≤ 1
The range of sin–1 x is the same as the restricted domain of
sin x :
– 2 ≤ sin–1 x ≤ 2
The inverse of cosine and
We can restrict the domains
of cos x and tan x in the same way
tangent
as we did for sin x so that
if
f(x) = cos x
then
f –1(x) = cos–1 x
And if
f(x) = tan x
then
f –1(x) = tan–1 x
for
for
for
for
0≤x≤π
–1 ≤ x ≤ 1.
– 2 < x < 2
x .
The graphs cos–1 x and tan–1 x can be obtained by reflecting
the graphs of cos x and tan x in the line y = x.
–1 x
The
graph
of
y
=
cos
y
y = cos x
–1


2
1
–1
0
–1
1

2

x
y = cosx
The domain of cos–1 x is the same as the range of cos x :
–1 ≤ x ≤ 1
The range of cos–1 x is the same as the restricted domain of
cos x :
0 ≤ cos–1 x ≤ π
–1 x
The graph yofy y= tan
=
tan
tanxx

y = tan–1 x
2
 2

x
2
 2
The domain of tan–1 x is the same as the range of tan x :
x
The range of tan–1 x is the same as the restricted domain of
tan x :
–  < tan–1 x < 
2
2
Problems involving inverse trig
Find the exact value of functions
sin
in radians.
–1
3
2
To solve this, remember the angles whose trigonometric ratios
can be written exactly:
radians
0

6

4

3

2
degrees
0°
30°
45°
60°
90°
sin
0
1
2
1
tan
0
3
2
1
3
3
2
1
2
1
cos
1
2
1
2
1
3
From this table
sin–1
3
2
= 
3
0

Problems involving inverse trig
Find the exact value of functions
sin 
in radians.
–1
2
2
This is equivalent to solving the trigonometric equation
cos θ = – 2
for 0 ≤ θ ≤ π
2
this is the range of cos–1x
We know that cos 4 = 1 = 22
2
Sketching y = cos θ for 0 ≤ θ ≤ π :
1
From the graph, cos 34 = – 22
2
2

0
2
2
–1


4
2
3
4

θ
So, cos–1 
2
2
= 3
4
Problems involving inverse trig
Find the exact value of functions
cos (sin
) in radians.
–1
Let sin–1
7
4
=θ
so
7
4
7
4
sin θ =
Using the following right-angled triangle:
7 + a2 = 16
a=3
4
7
θ
3
The length of the third side is 3 so
cos θ =
But sin–1
7
4
3
4
= θ so
cos (sin–1
7
)
4
=
3
4