The given figure represents the lines y = x +1 and y = 3 x -1. Write down the angles which the lines make with the positive direction of x-axis. Hence determine .
Notice where the lines cross the x-axis.
$\displaystyle y=x+1$ crosses at $\displaystyle x=-1$ and $\displaystyle \sqrt{3}x-1$ crosses at $\displaystyle \frac{1}{\sqrt{3}}$.
Since the lines form a triangle with the x-axis, you can find the lenght of the side opposite of $\displaystyle \theta$ using the x-intercepts.
So just find the point where the lines intersect by solving $\displaystyle x+1=\sqrt{3}x-1$. Once you find the point, you should be able to find the lenghts of the sides.
Hi snigdha,
The line whose equation is $\displaystyle y = x + 1$ has a slope of 1.
The acute angle formed with the positive direction of the x-axis (which is inside the triangle) is $\displaystyle \tan^{-1}(1)={\color{red}45^{\circ}}$
The line whose equation is $\displaystyle y=\sqrt{3}x-1$ has a slope of $\displaystyle \sqrt{3}$.
The acute angle formed with the positive direction of the x-axis is $\displaystyle \tan^{-1}(\sqrt{3})=60^{\circ}$.
Now we're interested in the obtuse angle inside the triangle, so we determine the supplement of $\displaystyle 60^{\circ}$ which is $\displaystyle {\color{red}120^{\circ}}$
Now, you have two angles of the triangle and can determine the value of $\displaystyle {\color{red}\Theta}$