Let a straight line l intersect the x - axis at A. The angle between the positive x - axis and the line l, measured in counter clockwise direction is called the angle of inclination of the straight line l.

In the above figure, If θ is the angle of a straight line l, then we have the following important points.

(i) 0° ≤ θ ≤ 180°

(ii) For horizontal lines, θ = 0° or 180° and for vertical lines, θ = 90°

(iii) If a straight line initially lies along the x-axis and starts rotating about a fixed point A on the x-axis in the counter clockwise direction and finally coincides with the x-axis, then the angle of inclination of the straight line in the initial position is 0°and that of the line in the final position is 0°.

(iv) Lines which are perpendicular to x-axis are called as vertical lines.

(v) Lines which are perpendicular to y-axis are called as horizontal lines.

(vi) Other lines which are neither perpendicular to x- axis and nor to y-axis are called as slant lines.

The major application of angle of inclination of a straight line is finding slope.

If θ is the angle of inclination of a straight line l, then tanθ is called the slope of gradient of the line is denoted by "m".

Therefore, the slope of the straight line is

**m = tan θ**

for 0° ≤ θ ≤ 180°

Let us find the slope of a straight using the above formula

(i) For horizontal lines, the angle of inclination is 0° or 180°.

That is,

θ = 0° or 180°

Therefore, slope of the straight line is

m = tan0° or tan 180° = 0

(ii) For vertical lines, the angle of inclination is 90°.

That is

θ = 90°

Therefore, slope of the straight line is

m = tan90° = Undefined

(iii) For slant lines, if θ is acute, then the slope is positive. Whereas if θ is obtuse, then the slope is negative.

When we look at a straight line visually, we can come to know the sign of the slope easily.

To know the sign of slope of a straight line, always we have to look at the straight line from left to right.

The figures given below illustrate this.

**Problem 1 :**

Find the angle of inclination of the straight line whose slope is 1/√3.

**Solution :**

Let θ be the angle of inclination of the line.

Then, slope of the line is

m = tanθ

**Given :** Slope = 1/√3

Then,

1/√3 = tanθ

θ = 30°

So, the angle of inclination is 30°.

**Problem 2 :**

If the angle of inclination of a straight line is 45°, find its slope.

**Solution :**

Let θ be the angle of inclination of the line.

Then, slope of the line,

m = tanθ

**Given :** θ = 45°

Then,

m = tan 45°

m = 1

So, the slope is 1.

**Problem 3 :**

If the angle of inclination of a straight line is 30°, find its slope.

**Solution :**

Let θ be the angle of inclination of the line.

Then, slope of the line,

m = tanθ

**Given :** θ = 30°

Then,

m = tan30°

m = 1/√3

So, the slope is 1/√3.

**Problem 4 :**

Find the angle of inclination of the straight line whose slope is √3.

**Solution :**

Let θ be the angle of inclination of the line.

Then, slope of the line,

m = tanθ

**Given :** Slope = √3

Then,

√3 = tanθ

θ = 60°

So, the angle of inclination is 60°.

**Problem 5 :**

Find the angle of inclination of the straight line whose equation is y = x + 32.

**Solution :**

Let θ be the angle of inclination of the line.

The given equation is in slope intercept form.

That is,

y = mx + b

Comparing

y = x + 32

and

y = mx + b,

we get the slope m = 1.

We know that the slope of the line is

m = tanθ

Then,

1 = tanθ

θ = 45°

So, the angle of inclination is 45°.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**