KONSEP DASAR TRIGONOMETRI YANG HARUS KAMU PAHAMI~Soal Penyelesaian Fisika dan Matematika

Matematika Untuk SMA - Rangkuman rumus dasar trigonometri yang harus kamu pahami karena rumus-rumus ini digunakan dalam berbagai soal matematika, kimia dan fisika, seperti turunan trigonometri, limit trigonometri, integral trigonometri, mekanika, listrik, dan sebagainya.

Perhatikan segitiga siku-siku berikut:

KONSEP DASAR TRIGONOMETRI YANG HARUS KAMU PAHAMI

Pada segitiga tersebut berlaku persamaan trigonometri sebagai berikut:

sin A = \frac{a}{c} cos A = \frac{b}{c} tan A = \frac{a}{b} csc A = \frac{1}{sin A} = \frac{c}{a} sec A = \frac{1}{cos α} = \frac{c}{B} cot A = \frac{1}{tan α} = \frac{b}{a}

$\small {\\\sin A =\frac {a} {c}\\\small \cos A =\frac{b} {c}\\\small \tan A =\frac {a}{b} \\\small \csc A =\frac{1}{\sin A }=\frac{c}{a} \\\small \sec A =\frac{1}{\cos \alpha }=\frac{c}{B} \\\small \cot A =\frac{1}{\tan \alpha }=\frac{b}{a} }$

Identitas Trigonomteri

tan α = \frac{sin α}{cos α} cot α = \frac{1}{tan α} = \frac{cos α}{sin α} sec α = \frac{1}{cos α} csc α = \frac{1}{sin α} {sin}^{2} α + {cos}^{2} α = 1 {tan}^{2} α + 1 = {sec}^{2} α {cot}^{2} α + 1 = {csc}^{2} α

$\small \\\tan \alpha = \frac{\sin \alpha }{\cos \alpha}\\\cot \alpha = \frac{1}{\tan \alpha}=\frac {\cos \alpha}{\sin \alpha}\\\sec \alpha =\frac {1}{\cos \alpha}\\\csc \alpha =\frac{1}{\sin \alpha}\\\sin ^2\alpha+\cos^2\alpha=1\\\tan ^2 \alpha +1=\sec^2 \alpha\\\cot^2\alpha+1=\csc^2\alpha$

Trigonometri penjumlahan dan pengurangan:

sin (α + β) = sin α cos β + cos α sin β sin (α - β) = sin α cos β - cos α sin β c o s (α + β) = cos α cos β - sin α sin β cos (α - β) = cos α cos β + sin α sin β tan (α + β) = \frac{tan α + tan β}{1 - tan α tan β} tan (α - β) = \frac{tan α - tan β}{1 + tan α tan β}

$\small \sin {(\alpha +\beta )}=\sin \alpha \cos \beta + \cos \alpha \sin \beta\\\small \sin {(\alpha -\beta )}=\sin \alpha \cos \beta - \cos \alpha \sin \beta\\\small cos {(\alpha +\beta )}=\cos \alpha \cos \beta -\sin \alpha \sin \beta\\\small \cos {(\alpha -\beta )}=\cos \alpha \cos \beta + \sin \alpha \sin \beta\\\small \tan (\alpha +\beta )=\frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta}\\\small \tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta}{1+\tan \alpha \tan \beta}$

Trigonometri Sudut Rangkap Dua:

$\small \\\sin 2\alpha =2 \sin \alpha \cos \alpha\\ \cos 2\alpha =\cos^2\alpha -\sin^2 \alpha\\\cos 2\alpha =2\cos^2\alpha -1\\\cos 2\alpha =1-2\sin^2\alpha \\\\ \tan 2\alpha =\frac{2\tan \alpha}{1-\tan^2 \alpha}\\\\\tan 2\alpha =\frac{2\cot \alpha }{\cot^2 \alpha -1}\\\\\tan 2\alpha =\frac{2}{\cot \alpha -\tan \alpha }$

Trigonometri Sudut Rangkap Tiga:

$\small \sin{3\alpha }=3\sin \alpha -4\sin^3 \alpha \\\small \cos{3\alpha }=4\cos^3 \alpha -3\cos \alpha$

Rumus trigonometri setengah sudut:

$\small \sin{\frac{1}{2}\alpha }=\pm\sqrt{\frac{1-\cos \alpha }{2}}\\\small \cos{\frac{1}{2}\alpha }=\pm\sqrt{\frac{1+\cos \alpha }{2}}\\\small \begin{align*}\tan{\frac{1}{2}\alpha }&=\pm\sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}\\&=\frac{\sin \alpha }{1+\cos \alpha }\\&=\frac{1-\cos \alpha }{\sin \alpha }\end{align*}$

Rumus Penjumlahan Trigonometri :

$\small \sin \alpha +\sin \beta=2\sin \frac{1}{2}(\alpha +\beta )\cos \frac{1}{2}(\alpha -\beta )\\\small \sin \alpha-\sin \beta=2\cos \frac{1}{2}(\alpha +\beta )\sin \frac{1}{2}(\alpha -\beta )\\\small \cos \alpha+\cos \beta=2\cos \frac{1}{2}(\alpha +\beta )\cos \frac{1}{2}(\alpha -\beta )\\\small \cos \alpha-\cos \beta=-2\sin \frac{1}{2}(\alpha +\beta )\sin \frac{1}{2}(\alpha -\beta )$

Rumus Perkalian Trigonometri:

$\small 2\sin \alpha \cos \alpha =\sin{(\alpha +\beta )}+\sin{(\alpha -\beta )}\\\small 2\cos \alpha \sin\beta =\sin{(\alpha +\beta )}-\sin{(\alpha -\beta )}\\\small 2\cos \alpha \cos\beta =\cos{(\alpha +\beta )}+\cos{(\alpha -\beta )}\\\small -2\sin \alpha \sin\beta =\cos{(\alpha +\beta )}-\cos{(\alpha -\beta )}$