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

Sep 21, 2016

{: (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
$\textcolor{w h i t e}{\text{XXX}} a = 10 , b = 20 , \angle C = {95}^{\circ}$

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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:
$\textcolor{w h i t e}{\text{XXX}} c = \sqrt{{10}^{2} + {20}^{2} - 2 \left(1\right) \left(20\right) \left(\cos \left({95}^{\circ}\right)\right)}$

$\textcolor{w h i t e}{\text{XXXX}} = 23.1$

By the Law of Sines
$\textcolor{w h i t e}{\text{XXX}} \sin \left(A\right) = \frac{\sin \left({95}^{\circ}\right) \times 10}{23.1} = 0.430748$

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

Similarly
$\textcolor{w h i t e}{\text{XXX}} \sin \left(B\right) = \frac{\sin \left({95}^{\circ}\right) \times 20}{23.1} = 0.861496$

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