November 2nd, 2018, 10:05 PM  #1 
Member Joined: Aug 2017 From: India Posts: 54 Thanks: 2  Angle conversion
I calculated arc tan or inverse tan of sqrt(3) and the result is 60. But when represented as 360 in the net it is shown as 120 Degrees. How 60 Deg is same as 120 Deg? Please advise. I know it will be something like 180  60 = 120. But why I should take 180 degrees and subtract 60? How that symmetry is coming?
Last edited by skipjack; November 2nd, 2018 at 11:45 PM. 
November 2nd, 2018, 10:53 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,584 Thanks: 1430 
$\begin{align*} &\tan(120^\circ) = \\ &\tan(60^\circ + 180^\circ) = \\ &\dfrac{\sin(60^\circ+180^\circ)}{\cos(60^\circ + 180^\circ)} = \\ &\dfrac{\sin(60^\circ)\cos(180^\circ)+\cos(60^\circ)\sin(180^\circ)} {\cos(60^\circ)\cos(180^\circ)\sin(60^\circ)\sin(180^\circ)} = \\ &\dfrac{\sin(60^\circ)}{\cos(60^\circ)} =\\ &\dfrac{\sin(60^\circ)}{\cos(60^\circ)} = \\ &\tan(60^\circ) \end{align*}$ 
November 2nd, 2018, 11:45 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259  
November 3rd, 2018, 09:05 AM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,584 Thanks: 1430  
November 3rd, 2018, 10:25 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
Perhaps, but I tried it on some websites and they got it right.

November 3rd, 2018, 10:32 AM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,584 Thanks: 1430 
Basically the idea is that that the tangent is the ratio of the sine and cosine functions and will remain the same if the signs of both of those are flipped. $\sin(\theta) = \sin(\theta + 180^\circ)$ $\cos(\theta) = \cos(\theta + 180^\circ)$ and thus $\tan(\theta) = \tan(\theta + 180^\circ)$ In order to convert from a domain of say $(180^\circ, 180^\circ]$ to $[0, 360)$ we simply add $180^\circ$ to the angle. In this case $60^\circ + 180^\circ = 120^\circ$ 
November 3rd, 2018, 10:43 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
It's easier to understand by considering the geometric definition of tan.

November 3rd, 2018, 05:18 PM  #8 
Member Joined: Aug 2017 From: India Posts: 54 Thanks: 2  
November 4th, 2018, 02:56 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259  ThetaCircle.PNG If, in the above diagram, x is nonzero, tan(θ) = y/x. If the point (x, y) is moved to a different position on the circumference of the circle, so that its coordinates become (x, y), the ratio of those coordinates is unchanged. The point's movement can be thought of as due to a rotation of the triangle through ±180$^\circ$, so that the angle θ becomes θ ± 180$^\circ$. The tangent of the angle is unchanged, i.e. y/x = tan(θ) = tan(θ ± 180$^\circ$). 

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