Appendix
A-7: Trigonometry
Example A-7.2
Two sides of a triangle have lengths a = 5 and b = 2 and the angle between them is C=52°. Use the Law of Sines and the Law of Cosines to find the other side c and the other two angles A and B opposite a and b, respectively.
Law of Sines:
sinAa=sinBb=sinCc
Law of Cosines:
c2=a2+b2−2 a b cosC
Solution
Interactive Solution
Use the Law of Cosines to find side c
Type the relevant equation.
Context Panel: Solve≻Numerically Solve Select the positive value.
c2=25+4−20 cos52⋅π180→solve−4.084944369,4.084944369
Use the Law of Sines to find angle A
Type the relevant equation, using the value of c computed above.
Context Panel: Solve≻Solve
5sinA=4.084944369sin52⋅π/180→solve1.303659266
Convert angle A to degrees
Context Panel: Approximate≻10 (digits)
180− 1.303659266⋅180/π→at 10 digits105.3058262
Because a2=25>b2+c2≐4+16≐20, angle A must fall in the second quadrant.
Coded Solution
Although the approach implemented above will work, the following variant, based on forming a sequence of three equations in A, B, c, each from the Law of Cosines, also works. The numeric solution of these equations will return angle A in the second quadrant.
Write a sequence of three equations in A,B,c
q≔c2=25+4−20 cos52⋅π180,25=c2+4−4 c cosA,4=25+c2−10 c cosB
Obtain a numeric solution via the fsolve command, imposing ranges on the variables.
fsolveq,A,B,c,A=0..π,B=0..π,c=0..5
A=1.837933386,B=0.3960880560,c=4.084944369
Use evalf to convert angle A to degrees.
evalf1.837933386⋅180/π
105.3058260
Apply the convert command with option degrees.
convert1.837933386,degrees
330.8280095degreesπ
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