Analytic geometry

02182019, 08:48 PM
(This post was last modified: 02192019 07:46 PM by Albert Chan.)
Post: #5




RE: Analytic geometry
(02182019 12:09 PM)yangyongkang Wrote: I will extend this question. Calculate the area enclosed by the three tangent lines of the circle. Area under rotated coordinate match area before rotation. So, again assume t1=0, we already know the base, find height: x cos(t2) + y sin(t2) = 1 ; rotated line 2 x cos(t3) + y sin(t3) = 1 ; rotated line 3 Multiply 1st eqn by sin(t3), 2nd by sin(t2), and subtract: x (cos(t2) sin(t3)  cos(t3) sin(t2)) = sin(t3)  sin(t2) x = (sin(t3)  sin(t2)) / sin(t3  t2) = cos(½(t3+t2)) sec(½(t3t2)) y value not needed here, but just in case: y = sin(½(t3+t2)) sec(½(t3t2)) height =  1  x  =  (cos(½(t3t2))  cos(½(t3+t2))) sec(½(t3t2))  =  2 sin(t2/2) sin(t3/2) sec(½(t3t2))  base =  tan(t2/2)  tan(t3/2)  =  (sin(t2/2) cos(t3/2)  cos(t3/2) sin(t2/2)) sec(t2/2) sec(t3/2)  =  sec(t2/2) sec(t3/2) sin(½(t3t2))  ΔArea = base*height/2, and remove the t1=0 restriction: ΔArea =  tan(½(t2t1)) tan(½(t3t1)) tan(½(t3t2))  

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Messages In This Thread 
Analytic geometry  yangyongkang  02182019, 09:27 AM
RE: Analytic geometry  yangyongkang  02182019, 12:09 PM
RE: Analytic geometry  Albert Chan  02182019 08:48 PM
RE: Analytic geometry  Albert Chan  02182019, 03:16 PM
RE: Analytic geometry  Albert Chan  02182019, 05:51 PM
RE: Analytic geometry  Albert Chan  02192019, 10:23 PM
RE: Analytic geometry  Albert Chan  02202019, 09:33 PM

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