**Angle of Inclination and Slope of a Line :**

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°.

After having gone through the stuff given above, we hope that the students would have understood how to do problems on angle of inclination and slope of a line.

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