hard maths questions please answer:

in the cartesian plan :

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

$\tan x=\frac{m_1-m_2}{1+m_1m_2}$

Ho Wei Haw - 7 years, 9 months ago

i did not understand the Penultimate equation

Abdou Ali - 7 years, 9 months ago

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

Daniel Liu - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$