33 Nonright Triangles

10. If the lengths of two sides are 14 and 12 and the included angle measures 23°, then what is the other side length and angles?

a=14;b=12;CC=23Degree;
c=N[(Sqrt[(a^2)+(b^2)-2a b Cos[CC]])]
AA=ArcCos[((c^2)+(b^2)-(a^2))/(2b c)]/Degree
BB=ArcCos[((c^2)+(a^2)-(b^2))/(2a c)]/Degree
AA=180-ArcSin[a*Sin[CC]/c]/Degree
BB=ArcSin[b*Sin[CC]/c]/Degree

CC

18. If two angles measure 95° and 22° and the length of the side between these two angles is 7, then what are the other side lengths?

AA=95Degree;BB=22Degree;CC=180Degree-AA-BB
c=7;
a=N[Sin[AA]*c/Sin[CC]]
b=N[Sin[BB]*c/Sin[CC]]

20. If the lengths of two sides are 8 and 17 and the angle opposite the length 17 side measures 98°, then what is the other side length and angles?

b=8;c=17;CC=98Degree;
a=aa/.Last[NSolve[(c^2)==(aa^2)+(b^2)-2aa b Cos[CC],aa]]
AA=ArcCos[((c^2)+(b^2)-(a^2))/(2b c)]/Degree
BB=ArcCos[((c^2)+(a^2)-(b^2))/(2a c)]/Degree

36. To measure the height of the Eiffel Tower in Paris, a person stands away from the base and measures the angle of elevation to the top of the tower to be 60°. Moving 210 feet closer, the angle of elevation to the top of the tower is 70°. How tall is the Eiffel Tower?

NSolve[{Tan[70Degree]==y/x,Tan[60Degree]==y/(x+210)},{x,y}]

37.  Two airplanes leave Kennedy airport in New York at 11 am. The air traffic controller reports that they are traveling away from each other at an angle of 103°.  The DC-10 travels 509 mph and the L-1011 travels at 503 mph. How far apart are they at 11:30 am?

a=509/2;b=503/2;CC=103Degree;
c=N[(Sqrt[(a^2)+(b^2)-2a b Cos[CC]])]