Pembezaan fungsi trigonometri

Trigonometri

Sejarah
Kegunaan
Fungsi
Fungsi songsang
Bacaan lanjut

Rujukan

Senarai identiti
Pemalar tepat
Menghasilkan jadual trigonometri

Teori Euclid

Hukum sinus
Hukum kosinus
Hukum tangen
Teorem Pythagoras

Kalkulus

Penggantian trigonometri
Kamiran fungsi
Pembezaan fungsi
Kamiran songsang

Pembezaan fungsi trigonometri merangkumi proses-proses matematik pembezaan kepada suatu fungsi trigonometri, yakni kadar perubahan berdasarkan suatu pemalar.

Semua hasil pembezaan bagi fungsi trigonometri boleh diterbitkan melalui proses pembezaan sin(x) dan kos(x). Peraturan hasil bahagi kemudiannya digunakan untuk menerbitkan hasil-hasil pembezaan lain.

Terbitan fungsi-fungsi trigonometri

Fungsi trigonometri asas

Terbitan pembezaan fungsi trigonometri asas
Fungsi Terbitan pembezaan
sin ( x ) {\displaystyle \sin(x)} d d x sin ( x ) = kos ( x ) {\displaystyle {d \over dx}\sin(x)=\operatorname {kos} (x)}
kos ( x ) {\displaystyle \operatorname {kos} (x)} d d x kos ( x ) = sin ( x ) {\displaystyle {d \over dx}\operatorname {kos} (x)=-\sin(x)}
tan ( x ) {\displaystyle \tan(x)} d d x tan ( x ) = d d x ( sin ( x ) kos ( x ) ) = kos 2 ( x ) + sin 2 ( x ) kos 2 ( x ) = 1 + tan 2 ( x ) = sek 2 ( x ) {\displaystyle {d \over dx}\tan(x)={d \over dx}{\biggl (}{\sin(x) \over \operatorname {kos} (x)}{\biggr )}={\operatorname {kos} ^{2}(x)+\sin ^{2}(x) \over \operatorname {kos} ^{2}(x)}=1+\tan ^{2}(x)=\operatorname {sek} ^{2}(x)}
sek ( x ) {\displaystyle \operatorname {sek} (x)} d d x sek ( x ) = d d x ( 1 kos ( x ) ) = sin ( x ) kos 2 ( x ) = sek ( x ) tan ( x ) {\displaystyle {d \over dx}\operatorname {sek} (x)={d \over dx}{\biggl (}{1 \over \operatorname {kos} (x)}{\biggr )}={\sin(x) \over \operatorname {kos} ^{2}(x)}=\operatorname {sek} (x)\tan(x)}
kosek ( x ) {\displaystyle \operatorname {kosek} (x)} d d x kosek ( x ) = d d x ( 1 sin ( x ) ) = kos ( x ) sin 2 ( x ) = kotan ( x ) kosek ( x ) {\displaystyle {d \over dx}\operatorname {kosek} (x)={d \over dx}{\biggl (}{1 \over \sin(x)}{\biggr )}=-{\operatorname {kos} (x) \over \sin ^{2}(x)}=-\operatorname {kotan} (x)\operatorname {kosek} (x)}
kotan ( x ) {\displaystyle \operatorname {kotan} (x)} d d x kotan ( x ) = d d x ( kos ( x ) sin ( x ) ) = kos ( x ) sin 2 sin 2 ( x ) = ( 1 + kot 2 ( x ) ) = kosek 2 ( x ) {\displaystyle {d \over dx}\operatorname {kotan} (x)={d \over dx}{\biggl (}{\operatorname {kos} (x) \over \sin(x)}{\biggr )}={-\operatorname {kos} (x)-\sin ^{2} \over \sin ^{2}(x)}=-(1+\operatorname {kot} ^{2}(x))=-\operatorname {kosek} ^{2}(x)}

Fungsi trigonometri songsang

Terbitan pembezaan fungsi trigonometri songsang
Fungsi Terbitan
arcsin ( x ) {\displaystyle \arcsin(x)} 1 1 x 2 {\displaystyle 1 \over {\sqrt {1-x^{2}}}}
arccos ( x c ) {\displaystyle \operatorname {arccos} (xc)} 1 1 x 2 {\displaystyle -1 \over {\sqrt {1-x^{2}}}}
arctan ( x ) {\displaystyle \arctan(x)} 1 1 + x 2 {\displaystyle 1 \over 1+x^{2}}
arcsec ( x ) {\displaystyle \operatorname {arcsec}(x)} 1 | x | 1 x 2 {\displaystyle 1 \over |x|{\sqrt {1-x^{2}}}}
arccsc ( x ) {\displaystyle \operatorname {arccsc}(x)} 1 | x | 1 x 2 {\displaystyle -1 \over |x|{\sqrt {1-x^{2}}}}
arccot ( x ) {\displaystyle \operatorname {arccot}(x)} 1 1 + x 2 {\displaystyle -1 \over 1+x^{2}}
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