Trigonometry pile up. How long is this side?

\( cos34^o=\frac{8}{a};\ cos34^o\approx0.829\\\\\frac{8}{a}=0.829\to a\approx9.65\\\\9.65-2.5+3.6=10.75\\\\cos11^o=\frac{b}{10.75};\ cos11^o\approx0.981\\\\\frac{b}{10.75}=0.981\to b\approx10.55\\\\10.55-2.1+2.9=11.35\\\\c^2+3.8^2=11.35^2\\\\c^2=11.35^2-3.8^2\\\\c\approx10.69 \)

\( 10.69-3.2=7.49\\\\tan42^o=\frac{d}{7.49};\ tan42^o\approx0.9\\\\\frac{d}{7.49}=0.9\to d\approx6.74\\\\sin37^o=\frac{6.74}{e};\ sin37^o\approx0.602\\\\\frac{6.74}{e}=0.602\to e\approx11.2\\\\11.2+4.3=15.5\\\\sin53^o=\frac{f}{15.5};\ sin53^o\approx0.8\\\\\frac{f}{15.5}=0.8\to f\approx12.4 \)

\( 12.4-2.2=10.2\\\\g^2=10.2^2+1.7^2\to g\approx10.34\\\\cos21^o=\frac{10.34}{h};\ cos21^o\approx0.934\\\\\frac{10.34}{h}=0.934\to h=11.07\\\\11.07+1.7=12.77\\\\sin71^o=\frac{x}{12.77};\ sin71^o\approx0.946\\\\\frac{x}{12.77}=0.946\to x\approx12.1\ (cm)\leftarrow Answer \)


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