Question

find the obtuse angle between lines having the slopes 1/2 and -1/3

Answer 1

Using Law of Cosines, angle is 135 degrees

From the slopes, draw a triangle connecting the 2 lines
use Pythagorean theorem to hypothesize possible lengths of triangle

Answer 2

can you explain it to me?

Answer 3

from the diagram, the angle between the horizontal and the line \(y=\frac{1}{2}x+c_2\) is \(a\), and the angle between the horizontal and the line \(y=-\frac{1}{3}x+c_1\) is \(b\).\[\tan(a)=\text{slope of line }y=\frac{1}{2}x+c_2\]so:\[\tan(a)=\frac{1}{2}\]similarly:\[\tan(b)=-\text{slope of line }y=-\frac{1}{3}x+c_1\]so:\[\tan(b)=\frac{1}{3}\]
you can then use the tan rule for a+b to get:\[\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\cdot\frac{1}{3}}=\frac{\frac{5}{6}}{\frac{5}{6}}=1\]therefore:\[a+b=45^0\]therefore the obtuse angle between the two lines is:\[180^0-(a+b)=180^0-45^0=135^0\]