Subjek:Matematika/Materi:Trigonometri

Hubungan fungsi trigonometri

Fungsi dasar:

$\sin A={\frac {a}{c}}\,$

$\cos A={\frac {b}{c}}\,$

$\tan A={\frac {\sin A}{\cos A}}\ ={\frac {a}{b}}\,$

$\cot A={\frac {1}{\tan A}}={\frac {\cos A}{\sin A}}\ ={\frac {b}{a}}\,$

$\sec A={\frac {1}{\cos A}}\ ={\frac {c}{b}}\,$

$\csc A={\frac {1}{\sin A}}\ ={\frac {c}{a}}\,$

$\sin {(-A)}=-\sin A$

$\cos {(-A)}=\cos A$

$\tan {(-A)}=-\tan A$

$\csc {(-A)}=-\csc A$

$\sec {(-A)}=\sec A$

$\cot {(-A)}=-\cot A$

Identitas trigonometri

$\sin ^{2}A+\cos ^{2}A=1\,$

$1+\tan ^{2}A={\frac {1}{\cos ^{2}A}}=\sec ^{2}A\,$

$1+\cot ^{2}A={\frac {1}{\sin ^{2}A}}=\csc ^{2}A\,$

Penjumlahan

$\sin(A+B)=\sin A\cos B+\cos A\sin B\,$

$\sin(A-B)=\sin A\cos B-\cos A\sin B\,$

$\cos(A+B)=\cos A\cos B-\sin A\sin B\,$

$\cos(A-B)=\cos A\cos B+\sin A\sin B\,$

$\tan(A+B)={\frac {\tan A+\tan B}{1-\tan A\tan B}}\,$

$\tan(A-B)={\frac {\tan A-\tan B}{1+\tan A\tan B}}\,$

$sin(u)+sin(v)=2sin({\frac {u+v}{2}})cos({\frac {u-v}{2}})$

$sin(u)-sin(v)=2cos({\frac {u+v}{2}})sin({\frac {u-v}{2}})$

$cos(u)+cos(v)=2cos({\frac {u+v}{2}})cos({\frac {u-v}{2}})$

$cos(u)-cos(v)=-2sin({\frac {u+v}{2}})sin({\frac {u-v}{2}})$

Perkalian

$2\sin A\times \cos B=\sin(A+B)+\sin(A-B),$

$2\cos A\times \sin B=\sin(A+B)-\sin(A-B),$

$2\cos A\times \cos B=\cos(A+B)+\cos(A-B),$

$2\sin A\times \sin B=-\cos(A+B)+\cos(A-B),$

Rumus sudut rangkap dua

$\sin 2A=2\sin A\cos A\,$

$\cos 2A=\cos ^{2}A-\sin ^{2}A=2\cos ^{2}A-1=1-2\sin ^{2}A\,$

$\tan 2A={2\tan A \over 1-\tan ^{2}A}={2\cot A \over \cot ^{2}A-1}={2 \over \cot A-\tan A}\,$

Rumus sudut rangkap tiga

$\sin 3A=3\sin A-4\sin ^{3}A\,$

$\cos 3A=4\cos ^{3}A-3\cos A\,$

Rumus setengah sudut

$\sin {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{2}}}\,$

$\cos {\frac {A}{2}}=\pm {\sqrt {\frac {1+\cos A}{2}}}\,$

$\tan {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{1+\cos A}}}={\frac {\sin A}{1+\cos A}}={\frac {1-\cos A}{\sin A}}\,$

Aturan Sinus, Cosinus, dan Tangen

Aturan sinus

${\frac {a}{\sin A}}\,=\,{\frac {b}{\sin B}}\,=\,{\frac {c}{\sin C}}\!$

Turunan dari aturan sinus

Luasan dari segitiga diatas dapat dirumuskan sebagai

$L={\frac {1}{2}}bc\sin A={\frac {1}{2}}ac\sin B={\frac {1}{2}}ab\sin C\,.$

Kalikan persamaan diatas dengan $2/abc$ maka akan menjadi

${\frac {2L}{abc}}={\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}\,.$

Aturan cosinus

$c^{2}=a^{2}+b^{2}-2ab\cos \gamma \ ,$

$a^{2}=b^{2}+c^{2}-2bc\cos \alpha \,$

$b^{2}=a^{2}+c^{2}-2ac\cos \beta \,$

Aturan tangen

${\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.$