h.mmss	RA		dec
Dubhe	11.034367	61.450372
Merak	11.015048	56.225673
Polaris	02.314909	89.155079

using linux calc
remark: computers use radians!
note: in this program "res" refers to the previous result; "?" means display result

convert to radians
* @hmmss.cal                  !this loads f(x) as the h.mmss->hour function
* let RAD=f(11.034367)*pi/12
* let decD=f(61.450372)*pi/180
* let RAM=f(11.015048)*pi/12
* let decM=f(56.225673)*pi/180
* let RAP=f(02.314909)*pi/12
* let decP=f(89.155079)*pi/180

RAD     =  2.896        DECD    =  1.078         RAM     =  2.888    
DECM    = 0.9841        RAP     = 0.6624         DECP    =  1.558    

find Dubhe-Merak separation:

* ? sin(decM)*sin(decD)+cos(decM)*cos(decD)*cos(RAM-RAD)
    .9956044985232644    
* ? acos(res)*180/pi
    5.374041950116754    
* let DM=res*pi/180

find Dubhe-Polaris separation:
* ? sin(decP)*sin(decD)+cos(decP)*cos(decD)*cos(RAP-RAD)
    .8770860112483803    
* ? acos(res)*180/pi
    28.70717600821549    
* let DP=res*pi/180

find Merak-Polaris separation:
* ? sin(decP)*sin(decM)+cos(decP)*cos(decM)*cos(RAP-RAM)
    .8283534790013846    
* ? acos(res)*180/pi
    34.07002986367971    
* let MP=res*pi/180

see: 28.70717/5.37404=5.34...so approx 5x

find the angle of the skinny triangle:
cos(C)=cos(A) cos(B) + sin(A) sin(B) cos(c)
cos(C)-cos(A) cos(B) = sin(A) sin(B) cos(c)
(cos(C)-cos(A) cos(B))/(sin(A) sin(B))= cos(c)

* ? (cos(MP)-cos(DP)*cos(DM))/(sin(DP)*sin(DM))
   -.9975680223503325    
* ? acos(res)*180/pi
    176.0032638800045 
i.e., a 4 deg deflection from a straight line

----
Swihard Ch.1, #1
* let RAsun=f(20.06)*pi/12
* let decsun=-20.3*pi/180
* let RAmoon=f(10.14)*pi/12
* let decmoon=15.8*pi/180
 
* ? sin(decsun)*sin(decmoon)+cos(decsun)*cos(decmoon)*cos(RAsun-RAmoon)
   -.8611756574128984    
* ? acos(res)*180/pi
    149.4488430096547    
remark: the angle between the Moon & Sun determines the phase of the Moon
since this angle >90 deg, this is a gibbous moon