正弦定理 - mtane0412

正弦定理

https://gyazo.com/9583d8875fa179c72488befaf2241ff9

なぜ使うか？：一辺とその両端の角から他の二辺が分かる

$ \frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R

辺を求める

$ a = \frac{b \ sinA}{sinB} = \frac{c \ sinA}{sinC}

$ b = \frac{a \ sinB}{sinA} = \frac{c \ sinB}{sinC}

$ c = \frac{a \ sinC}{sinA} = \frac{b \ sinC}{sinB}

角度を求める

$ sinA = \frac {a \ sinB}{b} = \frac {a \ sinC}{c} → $ \angle A = sin^{-1}(\frac {a \ sinB}{b}) = sin^{-1}(\frac {a \ sinC}{c})

$ sinB = \frac {b \ sinA}{a} = \frac {b \ sinC}{c} → $ \angle B = sin^{-1}(\frac {b \ sinA}{a}) = sin^{-1}(\frac {b \ sinC}{c})

$ sinC = \frac {c \ sinA}{a} = \frac {c \ sinB}{b} → $ \angle C = sin^{-1}(\frac {c \ sinA}{a}) = sin^{-1}(\frac {c \ sinB}{b})