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in the cartesian plan :

Note by Abdou Ali
3 years, 6 months ago

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I presume that $$x$$ refers to the angle between the two intersecting lines.

Firstly, notice that for every two intersecting lines in a Cartesian plane, there will be one line which makes a larger angle with the positive x-axis.

We shall call the gradient of this line as $$m_1$$. Similarly, we shall call the gradient of the line which makes a smaller angle with the positive x-axis as $$m_2$$

The relation between the angle and the gradient is given by

$m_1 = \tan \theta_1$

$m_2 = \tan \theta_2$

where $$\theta_1>\theta_2$$

Notice that $$x=\theta_1-\theta_2$$

So, taking tangents on both sides,

$\tan x=\tan(\theta_1-\theta_2)$

By the addition formula for tangent,

$\tan x=\frac{\tan\theta_1-\tan\theta_2}{1+\tan\theta_1\tan\theta_2}$

which is equivalent to

$\tan x=\frac{m_1-m_2}{1+m_1m_2}$ · 3 years, 6 months ago

i did not understand the Penultimate equation · 3 years, 6 months ago

It is the different of angles formula for tan. For example, $$\tan(a-b)=\dfrac{\tan a - \tan b}{1+\tan a \tan b}$$. · 3 years, 6 months ago