Thursday, April 21, 2011

Rumus-Rumus Trigonometri

Perhatikan Segitiga siku-siku ABC berikut:
segitiga

1. Sin \hspace{2mm}\alpha = \hspace{2mm}\frac {AC}{BC}
\vspace {10mm}
2. Cos \hspace{2mm}\alpha = \hspace{2mm}\frac {AB}{BC}
\vspace {10mm}
3. Tan \hspace{2mm}\alpha = \hspace{2mm}\frac {AC}{AB}
\vspace {10mm}
4. Ctg \hspace{2mm}\alpha = \hspace{2mm}\frac {AB}{AC}
\vspace {10mm}
5. Sec \hspace{2mm}\alpha = \hspace{2mm}\frac {BC}{AB}
\vspace {10mm}
6. Cosec \hspace{2mm}\alpha = \hspace{2mm}\frac {BC}{AC}

\vspace {10mm}
Rumus Identitas

1. Sin \sp{2}\alpha \hspace{2mm}+ cos \sp{2}\alpha \hspace {2mm}= 1
2. tan \sp{2}\alpha \hspace{2mm}+ 1 \hspace{2mm}=\hspace{2mm}Sec \sp{2}\alpha
3. Ctg \sp{2}\alpha \hspace{2mm}+ 1 \hspace{2mm}=\hspace{2mm}Cosec \sp{2} \alpha

\vspace {10mm}
Rumus Kebalikan

\vspace {10mm}
1. Cosec \alpha \hspace{2mm}=\hspace {2mm} \frac {1}{sin \alpha}
\vspace {10mm}
2. Sec \alpha \hspace{2mm}=\hspace {2mm} \frac {1}{Cos \alpha}
\vspace {10mm}
3. Ctg \alpha \hspace{2mm}=\hspace {2mm} \frac {1}{Tan \alpha}
Rumus Penjumlahan dan pengurangan

\sin(\alpha+\beta)=\sin \alpha \cos\beta + \cos\alpha \sin\beta
\sin(\alpha-\beta)=\sin \alpha \cos\beta -\cos\alpha \sin\beta
\cos(\alpha+\beta)=\cos \alpha \cos\beta - \sin\alpha \sin\beta
\cos(\alpha-\beta)=\cos \alpha \cos\beta +\sin\alpha \sin\beta
\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}
Rumus sudut rangkap

\sin(2\alpha)=2\sin\alpha\cos\alpha=\frac{2\tan\alpha}{1+\tan^{2}\alpha}
\cos(2\alpha)=\cos^{2}\alpha-\sin^{2}\alpha=1-2\sin^{2}\alpha=2\cos^{2}\alpha-1
\cos(2\alpha)=\frac{1-\tan^{2}\alpha}{1+\tan^{2}\alpha}
\tan(2\alpha)=\frac{2\tan\alpha}{1-\tan^{2}\alpha}
\tan(2\alpha)=\frac{2 \cot \alpha}{\cot^{2}\alpha-1}=\frac{2}{\cot \alpha - \tan\alpha}
Rumus setengah sudut

\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}
\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}
\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}=\frac{\sin\alpha}{1+\cos\alpha}
\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}

rumus untuk sudut


(3\alpha)
\sin(3\alpha)= 3\sin\alpha-4\sin^{3}\alpha
\cos(3\alpha)= 4\cos^{3}\alpha-3\cos\alpha




\tan(3\alpha)=\frac{3\tan\alpha-\tan^{3}\alpha}{1-3\tan^{2}\alpha}

Rumus perkalian

2\cos \alpha \cos \beta = \cos (\alpha+\beta)+\cos(\alpha-\beta)
2\sin \alpha \sin \beta = -\cos (\alpha+\beta)+\cos(\alpha-\beta)
2\sin \alpha \cos \beta = \sin (\alpha+\beta)+\sin(\alpha-\beta)
2\cos \alpha \sin \beta = \sin (\alpha+\beta)-\sin(\alpha-\beta)
\sin (\alpha + \beta).\sin(\alpha-\beta) = \sin^{2}\alpha-\sin^{2}\beta=\cos^{2}\beta-\cos^{2}\alpha
\cos (\alpha + \beta).\cos(\alpha-\beta) = \cos^{2}\alpha-\sin^{2}\beta=\cos^{2}\beta-\sin^{2}\alpha.

Rumus Jumlah dan selisih.

\cos \alpha +\cos \beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\cos \alpha -\cos \beta = -2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\sin \alpha +\sin \beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\sin \alpha -\sin \beta = 2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\tan \alpha +\tan \beta = \frac{\sin(\alpha+\beta)}{\cos\alpha.\cos\beta}
\tan \alpha -\tan \beta = \frac{\sin(\alpha-\beta)}{\cos\alpha.\cos\beta}.
Posted by at
Labels:

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)