Inner angles

The inner angles of the triangle are 30°, 45° and 105° and its longest side is 10 cm. Calculate the length of the shortest side, write the result in cm up to two decimal places.

Result

a =  5.18 cm

Solution:

A=30 B=45 C=105 c=10 cm a/c=sin⁡(A)/sin⁡(C)=sin⁡(30)/sin⁡(105)≐1.018 a=c⋅ sin⁡(A∘→ rad)/sin⁡(C∘→ rad)=c⋅ sin⁡(A∘⋅ π180 )/sin⁡(C∘⋅ π180 )=10⋅ sin⁡(30∘⋅ 3.1415926180 )/sin⁡(105∘⋅ 3.1415926180 )=5.17638=5.18 cmA=30 \ \\ B=45 \ \\ C=105 \ \\ c=10 \ \text{cm} \ \\ a/c=\sin(A)/\sin(C)=\sin(30)/\sin(105) \doteq 1.018 \ \\ a=c \cdot \ \sin( A ^\circ \rightarrow\ \text{rad})/\sin( C ^\circ \rightarrow\ \text{rad})=c \cdot \ \sin( A ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ )/\sin( C ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ )=10 \cdot \ \sin( 30 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ )/\sin( 105 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ )=5.17638=5.18 \ \text{cm}

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