Inclination of a line

The inclination of a line or angle of inclination is the acute or obtuse angle that is formed when a nonhorizontal line intersects the x-axis.

Formal definition:

The inclination of a nonhorizontal line is the positive angle θ with θ less than 180 degrees and measured counterclockwise from the x-axis to the line.


In the figure above, there are two angles. However, the angle in blue is the angle of inclination since it is measured counterclockwise according to the definition. We see that the slope is positive.

When the slope is positive, the angle of inclination is an acute angle.

In the figure above, you can again see that there are two angles. The angle in blue is once again the angle of inclination since it is measured counterclockwise according to the definition.

Notice though that this time the angle is bigger than 90 degrees and the slope is negative.

When the slope is negative, the angle of inclination is an obtuse angle.

Notice also that when a line intersects with the x-axis, 4 angles are formed as shown below.

However the angles in blue are vertical angles and therefore have the same measure. By the same token, the angles in red are vertical angles and also have the same measure. Therefore, we only need to concern ourselves with θ and θ

θ is not the angle of inclination, so we are left with
θ

Angle of inclination


The relationship between the slope of a line and its inclination is given by the following equation.

m = tan θ

How to derive the equation relating the inclination of a line to the slope (m = tan θ)


Slope is positive

Consider the following figure. Point (x1, 0) intersects the x-axis and (x2, y2) is another point on the line.


If m = 0, θ = 0 since the line is horizontal. Therefore, the result is true for horizontal lines since m = 0 = tan 0.

If the line has a positive slope, then

m =  
y2 - 0 / x2 - x1

m =  
y2 / x2 - x1
(Equation I)

tan θ =  
y2 / x2 - x1
(Equation II)


Combining equation I and equation II, we get m = tan θ

Slope is negative

Consider the following figure. Point (x1, 0) intersects the x-axis and (x2, y2) is another point on the line.

Derive angle of inclination when slope is negative
m =  
y2 - 0 / x2 - x1

m =  
y2 / x2 - x1
(Equation I)

Key concept: θ is the reference angle for θ
Therefore, sin(θ) = sin(θ)

for example, sin (120 degrees) = sin(60 degrees) = 0.8660254038

If sin(θ) = sin(θ), then tan(θ) = tan(θ)

tan(θ) = tan(θ) =  
y2 / x2 - x1
(Equation II)

m = tan θ


Exercises on how to find or calculate the angle of inclination


1) When the line is vertical, the inclination of a line is 90 degrees. In this case, there is no answer for the slope.

2) When the line is horizontal, the inclination of a line is 0 degree.

3) Find the slope when the inclination of a line is 45 degrees

Find the slope when the inclination of a line is 45 degrees


m = tan θ =  tan 45 degrees

Type 45 degrees and use the tan key on a scientific calculator. 

You will see that tan (45 degrees) = 1

m = 1

Now suppose, you are given a slope of 1 and want to find the angle of inclination.

θ = tan-1(1)

Type 1 and use the tan-1 key on a scientific calculator. 

You will see that θ = tan-1(1) = 45 degrees

4) Find the slope when the inclination of a line is 135 degrees

Find the slope when the inclination of a line is 135 degrees

m = tan θ = tan 135 degrees

Type 135 degrees and use the tan key on a scientific calculator. 

You will see that tan (135 degrees) = -1

m = -1

Now suppose, you are given a slope of -1 and want to find the angle of inclination.

Find tan-1(-1)

Type -1 and use the tan-1 key on a scientific calculator. 

You will see that tan-1(1) = -45 degrees

As you can see -45 degrees is not the 135 degrees angle we started with.

To get 135 degrees, you need to add 180 degrees to 45 degrees.

When the slope is negative, θ = 180 degrees + tan-1(-1) = 180 degrees + 45 degrees = 135 degrees

For more exercises, check this website




Recent Articles

  1. Distance between a Point and a Line - Formula, proof, and Examples

    Apr 23, 19 08:22 AM

    The easy way to find the distance between a point and a line. Learn how to derive the formula the easy way

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest


Math quizzes

 Recommended

Scientific Notation Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz

Types of angles quiz


Tough algebra word problems

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Recent Articles

  1. Distance between a Point and a Line - Formula, proof, and Examples

    Apr 23, 19 08:22 AM

    The easy way to find the distance between a point and a line. Learn how to derive the formula the easy way

    Read More

K-12 math tests


Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.