Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Functions » Trigonometric inequalities

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
Download
x

Download collection as:

  • PDF
  • EPUB (what's this?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(what's this?)" link.

  • More downloads ...

Download module as:

  • PDF
  • EPUB (what's this?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(what's this?)" link.

  • More downloads ...
Reuse / Edit
x

Collection:

Module:

Add to a lens
x

Add collection to:

Add module to:

Add to Favorites
x

Add collection to:

Add module to:

 

Trigonometric inequalities

Module by: Sunil Kumar Singh. E-mail the author

Evaluating trigonometric ratios is a direct process in which we make use of known values, trigonometric identities and transformations or even pre-defined trigonometric tables. The evaluation of trigonometric inequalities is somewhat inverse of this process. Consider an inequality :

tan x - 3 tan x - 3

Clearly, we need to know “x” for which this inequality holds. As pointed out earlier, trigonometric functions are “many-one” relation. The value of “x” satisfying given inequality is not an unique interval, but a series of intervals. Incidentally, however, trigonometric function values repeat after certain “period”. So this enables us to define periodic intervals in generic manner for which trigonometric inequality holds.

Figure 1: Intervals satisfying inequality, involving tangent function.
Trigonometric inequality
 Trigonometric inequality  (te1.gif)

We observe that line y = - 3 y = - 3 intersects tangent graph at multiple points. The sections of plots satisfying the inequality are easily identified on the graph and are shown as dark red line.

Solution of trigonometric inequality

Determination of base or fundamental interval is central to solve trigonometric inequality. The function values in this interval is repeated with a periodicity of trigonometric function. The base interval depends on the nature of trigonometric function and inequality in question. The steps to find solution of trigonometric inequality are :

1 : Convert given inequality to trigonometric equation by replacing inequality sign by equality sign.

2 : Solve resulting equation in the interval [0,2π]. There are two solutions. They are the angle values at which trigonometric function has the value which is being compared in the given inequality.

3 : Convert positive angle greater than π to equivalent negative value to account for the fact that basic interval being repeated may lie on negative side of the origin (cosine, secant and tangent function).

4 : Construct base interval between two values, keeping in mind the given inequality. It is always advantageous to draw a rough intersection of graphs of each side of given inequality.

5 : If function asymptotes (tangent, cotangent, secant and cosecant) within the interval constructed, then basic interval is limited by the angle value at which function asymptotes.

6 : Generalize solution by extending base interval with the period of the trigonometric function.

In order to understand the process, let us solve the inequality given by :

tan x - 3 tan x - 3

This example has been selected here as it involves consideration of each step as enumerated above for finding solution of inequality. Corresponding trigonometric equation, in this case, is :

tan x = - 3 tan x = - 3

The acute angle is π/3. Further, tangent function is negative in second and fourth quarter (see sign diagram). Using value diagram in conjunction with sign diagram, solution of given equation in [0, 2π] are :

Figure 2: Tangent function is negative in second and fourth quarter .
Sign and value diagram
 Sign and value diagram  (te2.gif)

x = π - θ = π - π 3 = 2 π 3 x = π - θ = π - π 3 = 2 π 3 x = 2 π - θ = 2 π - π 3 = 5 π 3 x = 2 π - θ = 2 π - π 3 = 5 π 3

Here, second angle is greater than π. Hence, equivalent negative angle is :

y = 5 π 3 - 2 π = - π 3 y = 5 π 3 - 2 π = - π 3

Tangent function, however, is not a continuous function between -π/3 and 2π/3. Tangent values are greater than -√3 for angle greater than -π/3, but value asymptotes to infinity at π/2. This can be verified from the intersection graph.

Figure 3: Intervals satisfying inequality, involving tangent function.
Trigonometric inequality
 Trigonometric inequality  (te1.gif)

Thus, basic interval satisfying inequality is :

- π 3 x < π 2 - π 3 x < π 2

It is also clear that the solution in this interval is repeated with a period of π, which is period of tangent function. Hence, solution of given inequality is :

n π - π 3 x < n π + π 2 ; n Z n π - π 3 x < n π + π 2 ; n Z

Examples

Example 1

Problem : Solve trigonometric inequality given by :

sin x 1 2 sin x 1 2

Solution : The solution of the corresponding equal equation is obtained as :

sin x = 1 2 = sin π 6 sin x = 1 2 = sin π 6

x = π 6 x = π 6

The sine function is positive in first and second quarter. Hence, second angle between “0” and “2π” is :

x = π θ = π π 6 = 5 π 6 x = π θ = π π 6 = 5 π 6

Both angles are less than “π”. Thus, we do not need to convert angle into equivalent negative angle. Further, sine curve is defined for all values of “x”. The base interval, therefore, is :

The valid intervals on sine plot are shown in the figure.

Figure 4: Intervals satisfying inequality
Trigonometric inequality
 Trigonometric inequality  (te4.gif)

π 6 x 5 π 6 π 6 x 5 π 6

The periodicity of sine function is “2π”. Hence, we add “2nπ” on either side of the base interval :

2 n π + π 6 x 2 n π + 5 π 6 , n Z 2 n π + π 6 x 2 n π + 5 π 6 , n Z

Example 2

Problem : Solve trigonometric inequality given by :

sin x > cos x sin x > cos x

Solution : In order to solve this inequality, it is required to convert it in terms of inequality of a single trigonometric function.

sin x > cos x sin x > cos x

sin x cos x > 0 sin x cos x > 0

sin x cos π 4 cos x sin π 4 > 0 sin x cos π 4 cos x sin π 4 > 0

sin x π 4 > 0 sin x π 4 > 0

Let y = x π / 4 y = x π / 4 . Then,

sin y > 0 sin y > 0

Thus, we see that problem finally reduces to solving trigonometric sine inequality. The solution of the corresponding equality is obtained as :

sin y = 0 = sin 0 sin y = 0 = sin 0

y = 0 y = 0

The second angle between “0” and “2π” is “π”. The base interval, therefore, is :

0 < y < π 0 < y < π

The periodicity of sine function is “2π”. Hence, we add “2nπ” on either side of the base interval :

2 n π < y < 2 n π + π , n Z 2 n π < y < 2 n π + π , n Z

Now substituting for y = x π / 4 y = x π / 4 , we have :

2 n π < x π 4 < 2 n π + π , n Z 2 n π < x π 4 < 2 n π + π , n Z

2 n π + π 4 < x < 2 n π + 5 π 4 , n Z 2 n π + π 4 < x < 2 n π + 5 π 4 , n Z

Example 3

Problem : If domain of a function, “f(x)”, is [0,1], then find the domain of the function given by :

f 2 sin x 1 f 2 sin x 1

Solution : The domain of the function is given here. We need to find the domain when argument (input) to the function is a trigonometric expression. The given domain is :

0 x 1 0 x 1

Changing argument of the function, the domain becomes :

0 2 sin x 1 1 1 2 sin x 2 1 / 2 sin x 1 0 2 sin x 1 1 1 2 sin x 2 1 / 2 sin x 1

However, the range of sinx is [-1,1]. It means that the above interval is equivalent to a trigonometric inequality given by :

sin x 1 2 sin x 1 2

The sine function is positive in first and second quadrant. Two values of “x” between “0” and “2π” are :

π 6 , π π 6 π 6 , π π 6

π 6 , 5 π 6 π 6 , 5 π 6

The value of “x” satisfying above equation :

2 n π + π / 6 < = x < = 2 n π + 5 π / 6, n Z 2 n π + π / 6 < = x < = 2 n π + 5 π / 6, n Z

Hence, required domain is :

[ 2 n π + π 6 , 2 n π + 5 π 6 ] , n Z [ 2 n π + π 6 , 2 n π + 5 π 6 ] , n Z

Example 4

Problem : Find the domain of the function given by :

f x = log e 1 [ cos x ] [ sin x ] f x = log e 1 [ cos x ] [ sin x ]

Solution : The function is a logarithmic function, which is valid for all positive values of its argument. Also, the argument of logarithmic function is in rational form, having denominator as a square root. We have to find values of “x” for which the expression within the square root is a positive number. It means that :

[ cos x ] [ sin x ] > 0 [ cos x ] [ sin x ] > 0

[ cos x ] > [ sin x ] [ cos x ] > [ sin x ]

In order to evaluate this inequality, we determine modulus of two trigonometric functions in four quadrants at all bounding values of angle “x” and also at intermediate angles. The values are shown in the figure :

Figure 5: The values of modulus of sine and cosine functions in four quadrants are shown.
Modulus of trigonometric functions
 Modulus of trigonometric functions   (t3a.gif)

It is clear that [ cos x ] > [ sin x ] [ cos x ] > [ sin x ] is true in the fourth quadrant. Hence, domain of the function is :

Domain = π 2 x 0 = [ - π 2, 0 ] Domain = π 2 x 0 = [ - π 2, 0 ]

Exercise

Exercise 1

Solve the inequality :

tan x < 1 tan x < 1

Exercise 2

Find domain of function :

y = log e cos x y = log e cos x

Exercise 3

Find domain of function :

f(x) = sin x - 1 f(x) = sin x - 1

Exercise 4

Find domain of function :

f x = 1 sin x + 3 sin x f x = 1 sin x + 3 sin x

Exercise 5

A function f(x) is defined in [0,1]. Determine range of function definition f(sinx).

Exercise 6

A function f(x) is defined in [0,1]. Determine range of function definition f(tanx).

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Reuse / Edit:

Reuse or edit collection (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.

| Reuse or edit module (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.