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.

Correct result:

a =  5.18 cm

Solution:

$$\begin{gathered} A=30 \\ B=45 \\ C=105 \\ c=10\text{ cm } \\ a\mathrm{/}c=\sin (A)\mathrm{/}\sin (C)=\sin (30)\mathrm{/}\sin (105)\doteq 1.018 \\ a=c\cdot \sin A^{\circ }\mathrm{/}\sin C^{\circ }=c\cdot \sin 30^{\circ }\mathrm{/}\sin 105^{\circ }=10\cdot \sin 30^{\circ }\mathrm{/}\sin 105^{\circ }=10\cdot 0.5\mathrm{/}0.965926=5.18=5.18\text{ cm} \end{gathered}$$

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