Question

Why is tan θ = slope of line ?

Answer 1

Why is $\tan \theta =$ slope of line?

Answer:

To explain why is $\tan \theta =$ slope of line we need to define the slope. The slope of a line is a number that measures its "steepness", usually denoted by the letter $m$ . It is the change in $y$ for a unit change in $x$ along the line. The slope of a line (also called the gradient of a line) is a number that describes how "steep" it is. Also we know one of the most important properties of a straight line is in how it angles away from the horizontal. This concept is reflected in something called the "slope" of the line.

Consider a straight line that passes through the points $\mathbf{A}(\mathbf{x}_1\mathbf{y}_1)$ and $\mathbf{B}(\mathbf{x}_2\mathbf{y}_2)$ . We know that the gradient, $m$ , of the line is defined by:

$$m = \frac {\text {c h a n g e i n y}}{\text {c h a n g e i n x}} = \frac {B C}{A C} = \frac {y _ {2} - y _ {1}}{x _ {2} - x _ {1}} \quad x _ {2} \neq x _ {1} (\text {F r o m t h e g r a p h b e l o w})$$

Where, $m =$ slope of the line.

Let $\theta$ be the angle (inclination angle). Then in the right angled triangle, Perpendicular = $y_{2} - y_{1}$ and base $= x_{2} - x_{1}$

$$m = \frac {\text {c h a n g e i n y}}{\text {c h a n g e i n x}} = \frac {B C}{A C} (\frac {\text {P e r p e n d i c u l a r}}{\text {B a s e}})$$

Therefore, $\tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$

From the definition of the line and tan $\theta$ we get that slope of the line $= \tan \theta$

$$m = \tan \theta \text {slope of the line}$$