Basic coordinate geometry formula to find the angle between two lines

• Dec 10th 2012, 08:41 AM
kinhew93
Basic coordinate geometry formula to find the angle between two lines
I just read that the angle between two lines with gradients m' and m'' can be found by:

tan(angle) = (m'' - m')/(a+(m'm''))

In try to prove this I saw that the angle can also be found by:

angle = invtan(m'') - invtan(m') which seems much simpler

Is this correct and if so why would we bother with the first formula?
• Dec 10th 2012, 09:39 AM
ebaines
Re: Basic coordinate geometry formula to find the angle between two lines
Quote:

Originally Posted by kinhew93
I just read that the angle between two lines with gradients m' and m'' can be found by:

tan(angle) = (m'' - m')/(a+(m'm''))

I think what you meant to write is this:
$\tan(a) = \frac {m''-m'}{1 + m'm''}$

Hence:
$a = arctan(\frac {m''-m'}{1 + m'm''} )$

And as yuo point out you can also use:
$a = arctan(m') - arctan(m'')$

Both equations are equivalent. Use whichever you like, but I agree that the second form is more intuitively obvious.
• Dec 10th 2012, 11:05 AM
Plato
Re: Basic coordinate geometry formula to find the angle between two lines
Quote:

Originally Posted by kinhew93
I just read that the angle between two lines with gradients m' and m'' can be found by:
tan(angle) = (m'' - m')/(a+(m'm''))
In try to prove this I saw that the angle can also be found by:
angle = invtan(m'') - invtan(m') which seems much simpler
Is this correct and if so why would we bother with the first formula?

Consider two simple lines: $y=3x~\&~y=-5x$.

Now test your conclusion on those lines.

What do you get?