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# The law of sines and examples

The law of sines is very useful for solving various tasks with not right triangles.

The law of sines works for any triangle. The only important thing is that side a faces angle A, side b faces angle B and side c faces angle C. It helps for correct using of formulas.

$$\frac{\text{a}}{\sin\text{A}}=\frac{\text{b}}{\sin\text{B}}=\frac{\text{c}}{\sin\text{C}}$$ Triangle. A, B, C are angles and a, b, c are sides.

## Solved examples

For triangle below find the side a. Round your answer to the nearest tenth. $$\frac{\text{a}}{\sin\text{45°}}=\require{cancel} \cancel{\frac{\text{b}}{\sin\text{B}}}=\frac{\text{40}}{\sin\text{35°}}$$ $$a=\frac{\text{40}}{\sin\text{35°}}\cdot \sin\text{45°} \approx 39.2$$

For triangle below find the angle A. Round your answer to the nearest one. $$\frac{\text{18}}{\sin\text{A}}=\frac{\text{23}}{\sin\text{75°}}=\require{cancel} \cancel{\frac{\text{c}}{\sin\text{C}}}$$ $$\sin\text{A}=\frac{18\cdot \sin\text{75°}}{23}= 0.7559...$$ $$\angle \text{A}=\sin^{-1}(0.7559...)\approx 49°$$