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Find the Equation of line which makes an angle of $\frac {π}{6}$ with the positive $x$-axis and passes through $(4,-4)$.

Hi guys, I have this problem which I don't know where should I start. Hope you guys can give a head start. I know the equation must be in $$y=mx+b$$ form and we have an $x,y$ intercept which is $(4,-4)$.

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  • $\begingroup$ You were given an angle of inclination and a point on the line. The point $(4, -4)$ is neither an $x$-intercept nor a $y$-intercept since it does not lie on either axis. $\endgroup$ – N. F. Taussig Jan 20 '16 at 13:02
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So the slope of the line is $m=\tan\left(\frac{\pi}{6}\right)$. Now the equation of the line is given by $$(y-y_{1}) = m \cdot (x-x_{1})$$ where your $(x_{1},y_{1})=(4,-4)$. Substitute and get the answer.

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  • $\begingroup$ Thanks a lot. I did the same and Got the Answer $\endgroup$ – Rahul Jan 20 '16 at 4:17
  • $\begingroup$ @user306498 No problem. Glad to help. Kindly accept an aswer $\endgroup$ – crskhr Jan 20 '16 at 4:18
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To make an angle of pi/6 it becomes trivial if you know your radian circle.

pi/6 refers to the angle that makes the triangle with rise of 1, and run of root(3). Perhaps this is enough to help you finish.

I would strongly recommend learning the radian circle if you intend to continue with calculus.

There are two important right triangles: one with sides 1, 1 and hypotenuse root(2). And the one that will help you with this question: sides 1, root(3), and hypotenuse of 2.

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By using the equation given which is $$y=mx+b$$ Note that $m=\tan (\pi/6)=1/\surd3$.
Then we get $y=(1/\surd3)x+b$
Substitute $(4,-4)$, we get $-4=(1/\surd3)(4)+b$ which implies that $b=-4-4/\surd3$
Hence the equation is $$y=(1/\surd 3)x+(-4-4/\surd3)$$

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