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

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

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

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

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

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