November 10th, 2016, 03:13 AM  #1 
Newbie Joined: Nov 2016 From: Uk Posts: 11 Thanks: 0  Arctan identity
Hey I'm trying to answer this without a calculator. Arctan(2+sqrt(3)) with calculator it is 75. Last edited by skipjack; November 10th, 2016 at 11:09 PM. 
November 10th, 2016, 03:31 AM  #2 
Senior Member Joined: Sep 2007 From: USA Posts: 349 Thanks: 67 Math Focus: Calculus 
I assume that is in degrees. You can reference a table of the unit circle to see that is the case. If you don't have a table that precise, use the halfangle formula of tangent for 150 degrees.
Last edited by Compendium; November 10th, 2016 at 03:33 AM. 
November 10th, 2016, 04:03 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,140 Thanks: 1415 
tan(75°) = tan(45° + 30°) = (tan(45°) + tan(30°))/(1  tan(45°)tan(30°)). As tan(45°) = 1 and tan(30°)= 1/√3, tan(75°) = (√3 + 1)/(√3  1) = 2 + √3. 
November 10th, 2016, 04:28 AM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,656 Thanks: 1327 
$0 < t < 90$ $\tan{t}=2+\sqrt{3} \implies \sin{t}=2\cos{t}+\sqrt{3}\cos{t}$ $2\cos{t}=\sin{t}\sqrt{3}\cos{t}$ $\cos{t}=\dfrac{1}{2}\sin{t}\dfrac{\sqrt{3}}{2}\cos{t}$ $\cos{t}=\sin(150)\sin{t}+\cos(150)\cos{t}$ $\cos{t}=\cos(150t) \implies t=150t \implies t = 75$ 
November 10th, 2016, 10:09 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 18,140 Thanks: 1415  arctan.png arctan(2 + √3) = 60° + 15° = 75°. 
November 11th, 2016, 07:49 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,637 Thanks: 959 Math Focus: Elementary mathematics and beyond 
$$2+\sqrt3=\frac{\sin60+\sin90}{\sin150}= \frac{2\sin^275}{2\cos75\sin75}=\tan75\implies \arctan(2+ \sqrt3)=75^\circ$$
Last edited by greg1313; November 11th, 2016 at 07:53 PM. 
November 11th, 2016, 11:05 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,140 Thanks: 1415 
Using cot(A) = csc(2A) + cot(2A), 2 + √3 = csc(30°) + cot(30°) = cot(15°) = tan(75°).

November 12th, 2016, 05:26 AM  #8 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,658 Thanks: 651 Math Focus: Wibbly wobbly timeywimey stuff. 
Wow! The solutions just keep on coming! Dan 

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