Question

Given a `triangleABC` with `a=10`, `b=20` and `/_C=95^circ`, what are the lengths of all sides and size of all angles?

Answer 1

#{: (a,color(white)("XX"),b,color(white)("XX"),c,color(white)("XX"),/_A,color(white)("XX"),/_B,color(white)("XX"),/_C), (10,color(white)("XX"),20,color(white)("XX"),23.1,color(white)("XX"),25.5^@,color(white)("XX"),59.5^@,color(white)("XX"),95^@) :}#

Explanation:

Use the Law of Cosines and the Law of Sines:

Given
`color(white)("XXX")a=10, b=20, /_C=95^@`

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A calculator/spreadsheet will be used for all calculations beyond this point and all final values should be taken as accurate to 1 decimal digit).
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By the Law of Cosines:
`color(white)("XXX")c=sqrt(10^2+20^2-2(1)(20)(cos(95^@)))`

`color(white)("XXXX")=23.1`

By the Law of Sines
`color(white)("XXX")sin(A)=(sin(95^@)xx10)/23.1=0.430748`

`color(white)("XXX")A = "arcsin"(sin(A))="arcsin"(0.430748)=25.5^@`

Similarly
`color(white)("XXX")sin(B)=(sin(95^@)xx20)/(23.1)=0.861496`

`color(white)("XXX")B="arcsin"(sin(B))="arcsin"(0.861496)=59.5^@`