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Kamis, 28 Juli 2016

On 14.59.00 by Kevin AS   No comments

Hubungan fungsi trigonometri


TrigonometryTriangle.svg
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}

Tabel Trigonometri



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

Rumus jumlah dan selisih sudut


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}

Rumus perkalian trigonometri


2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A sin B = sin (A + B) - sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
2 sin A sin B = - cos (A + B) + cos (A - B)

Rumus jumlah dan selisih trigonometri


sin A + sin B = 2 sin \frac{1}{2} (A + B) cos \frac{1}{2} (A - B)
sin A - sin B = 2 cos \frac{1}{2} (A + B) sin \frac{1}{2} (A - B)
cos A + cos B = 2 cos \frac{1}{2} (A + B) cos \frac{1}{2} (A - B)
cos A -  cos B = - 2 sin \frac{1}{2} (A + B) sin \frac{1}{2} (A - B)

Rumus sudut rangkap dua


sin 2A = 2 sin A cos A
cos 2A = cos^2 A - sin^2 A   
            = 1 - 2 sin^2 A 
            = 2 cos^2 A - 1
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+cosA}} = \frac {sin A}{1+cos A} = \frac {1-cos A}{sin A}

Turunan Fungsi Trigonometri


{\displaystyle (\sin x)'=\cos x\,}{\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}
{\displaystyle (\cos x)'=-\sin x\,} {\displaystyle (\arccos x)'={-1 \over {\sqrt {1-x^{2}}}}\,}
{\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}\,} {\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}
{\displaystyle (\sec x)'=\sec x\tan x\,} {\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}
{\displaystyle (\csc x)'=-\csc x\cot x\,} {\displaystyle (\operatorname {arccsc} x)'={-1 \over |x|{\sqrt {x^{2}-1}}}\,}
{\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}\,} {\displaystyle (\operatorname {arccot} x)'={-1 \over 1+x^{2}}\,}

Integral Trigonometri


{\displaystyle \int \sin x\,dx=-\cos x+C\,}
{\displaystyle \int \cos x\,dx=\sin x+C\,}
{\displaystyle \int \tan x\,dx=\ln |\sec x|+C\,}
{\displaystyle \int \cot x\,dx=\ln |\sin x|+C\,}
{\displaystyle \int \sec x\,dx=\ln |\sec x+\tan x|+C\,}
{\displaystyle \int \csc x\,dx=\ln |\csc x-\cot x|+C\,}
{\displaystyle \int \sec ^{2}x\,dx=\tan x+C\,}
{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C\,}
{\displaystyle \int \sec x\tan x\,dx=\sec x+C\,}
{\displaystyle \int \csc x\cot x\,dx=-\csc x+C\,}