My Math Forum Triangle inside another problem

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 June 14th, 2016, 01:27 PM #1 Newbie   Joined: Jun 2016 From: home Posts: 1 Thanks: 0 Triangle inside another problem Hi I have a problem which I am sure is simple, but I can't get it right. Given is beta, x, and y. Required is Phi. Can you help? Last edited by skipjack; June 14th, 2016 at 01:43 PM.
 June 14th, 2016, 02:55 PM #2 Global Moderator   Joined: May 2007 Posts: 6,806 Thanks: 716 Notation: u=long hypotenuse, v=short hypotenuse, w=vertical side of triangles. b=beta, p=phi 1)y/u=sinb or u=y/sinb. 2)w=ucosb. 3)(y-x)/w=tan(b-p). 4)p=b-arctan(b-p)=b-arctan((y-x)/w). Carry out steps above in order.
 June 14th, 2016, 04:07 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond Let $h$ be the height of the triangle. Then $$h=\dfrac{y}{\tan\beta},\quad\phi=\beta-\tan^{-1}\left(\dfrac{y-x}{h}\right)$$
 June 14th, 2016, 04:25 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,927 Thanks: 2205 $\phi$ = tan$^{-1}$(sin(β)cos(β)/(y/x - sin²(β))) by mathman's method, but greg1313's way of writing it is simpler and implies $\phi = \beta - \tan^{-1}((1 - x/y)\tan\beta)$.

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