Question

Solve triangle ABC which haveangleC=1250:431;a=4:2cmandc=8:2cm:Findb:

Answer 1

Answer on Question #51012 – Math – Trigonometry

Task

Solve triangle ABC which have angle $C = 125{}^\circ$; $a = 4\,\mathrm{cm}$ and $c = 8\,\mathrm{cm}$. Find b.

Solution

According to the Sine Rule,

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \Rightarrow \frac{a}{\sin A} = \frac{c}{\sin C} \Rightarrow \sin A = \frac{a \sin C}{c} = \frac{4 \sin (125{}^\circ)}{8} = \frac{\sin (125{}^\circ)}{2} \Rightarrow$$ $$A = \arcsin \frac{\sin (125{}^\circ)}{2} \approx 24.2{}^\circ$$ $$A + B + C = 180{}^\circ \Rightarrow B = 180{}^\circ - A - C = 30.8{}^\circ$$ $$\frac{a}{\sin A} = \frac{b}{\sin B} \Rightarrow b = \frac{a \sin B}{\sin A} = \frac{4 \sin (30.8{}^\circ)}{\sin (125{}^\circ)} \approx 2.5\,\mathrm{cm}$$

Answer: $b = 2.5\,\mathrm{cm}$

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