[Math] Calculate all Angles of the Triangle (all 3 sides given)

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so id like to calculate the 3 given Angles as efficient as possible with trigometrics. Maybe with the law of Cosine. So if someone knows the formula for this id be happy to know.

Thanks for help

Answer 1

The largest angle is opposite the longest side. To find the largest angle, use the Law of Cosines $$c^2 = a^2 + b^2 - 2ab\cos\gamma$$ with $a = 7~\text{cm}$, $b = 5~\text{cm}$, and $c = 10~\text{cm}$.

Once you have solved for $\gamma$, use the Law of Sines $$\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}$$ to determine $\alpha$. Keep in mind that there is only one possible value for $\alpha$ since $\alpha < \gamma$.

Once you have determined both $\alpha$ and $\gamma$, you can obtain $\beta$ by using the Angle Sum Theorem for Triangles, $\alpha + \beta + \gamma = 180^\circ$.