Understanding the Domains and Ranges of Trigonometric Functions, Study notes of Mathematics

An explanation of the one-to-one relationships between trigonometric functions such as sin, cos, and tan, along with their domains and ranges. It emphasizes the importance of checking if the inverse functions are defined for the given values and if the angles fall within the legal limits.

Typology: Study notes

Pre 2010

Uploaded on 07/22/2009

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xy sin
(One to one)
xy
1
sin
Domain
2
,
2
1,1
Range
1,1
2
,
2
Now look at
1)
xx
)(sinsin
1
if
x
2
,
2
2)
xx
)sin(sin
1
if
x
1,1
xy cos
(One to one)
Domain
,0
1,1
Range
1,1
,0
Again,
3)
xx
)(coscos
1
if
x
,0
4)
xx
)cos(cos
1
if
x
1,1
xy tan
(One to one)
xy
1
tan
Domain
2
,
2
,
Range
,
2
,
2
Now look at
5)
xx
)(tantan
1
if
x
2
,
2
6)
xx
)tan(tan
1
if
x
,
In both cases x must
belong to the range of
the outer function.
Here also in both cases
x must belong to the
range of the outer
function.
Once again, in both
cases x must belong to
the range of the outer
function.
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y sin x (One to one) y x

1 sin

 

Domain 

 

Range ^ ^1 ,^1 

Now look at

1) x  x

 sin (sin )

1

if x^ 

 

2) x^  x

 sin(sin )

1

if x^ ^ ^1 ,^1 

y cos x

(One to one) y^ x

1 cos

 

Domain ^0 ,^  ^ ^1 ,^1 

Range ^ ^1 ,^1  ^0 ,^ 

Again,

3) x^  x

 cos (cos )

1

if x^ ^0 ,^ 

4) x  x

 cos(cos )

1

if x^  ^ ^1 ,^1 

y  tan x (One to one) y x

1 tan

 

Domain 

 

Range ^ ^ ,^  

Now look at

5) x^  x

 tan (tan )

1

if x^ 

 

6) x  x

 tan(tan )

1

if x^  ^ ^ ,

In both cases x must

belong to the range of

the outer function.

Here also in both cases

x must belong to the

range of the outer

function.

Once again, in both

cases x must belong to

the range of the outer

function.

 When you try to find the value of a problem such as )

sin(cos

 1

you must pay attention to two different things.

1) Is the inverse part of the problem defined for the indicated value?

In this problem the question is

1 cos

defined for

? If it is defined, then the next

concern should be that

2) The angle that corresponds to

cos

 1

must fall within the rage of

1 cos

which is ^0 ,^ .

So when you do the problem as explained in the class that “findsin  when cos 

” make sure that the  is with in the legal limits.

I thought that we had a good class today. I enjoyed trying to answering your

questions.