User:P1ayer/備頁/三角函數轉換表

基本函數

當平面上的三點O、C、A的連線， ${\overline {OC}}$ 、 ${\overline {CA}}$ 、 ${\overline {OA}}$ ，構成一個直角三角形。對於 ${\overline {OC}}$ 與 ${\overline {OA}}$ 的夾角 ${\angle \theta }$ 而言：

對邊（opposite）

y={\overline {CA}}

鄰邊（adjacent）

x={\overline {OC}}

斜邊（hypotenuse）

r={\overline {OA}}

函数	英语	简写	定義	关系
正弦	Sine	sin	${\frac {y}{r}}$	$\sin \theta ={\frac {1}{\csc \theta }}$
餘弦	Cosine	cos	${\frac {x}{r}}$	$\cos \theta ={\frac {1}{\sec \theta }}\,$
正切	Tangent	tan	${\frac {y}{x}}$	$\tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {1}{\cot \theta }}$
餘切	Cotangent	cot	${\frac {x}{y}}$	$\cot \theta ={\frac {\cos \theta }{\sin \theta }}={\frac {1}{\tan \theta }}$
正割	Secant	sec	${\frac {r}{x}}$	$\sec \theta ={\frac {1}{\cos \theta }}$
餘割	Cosecant	csc	${\frac {r}{y}}$	$\csc \theta ={\frac {1}{\sin \theta }}$

基本關係

函數	sin	cos	tan	cot	sec	csc
$\sin \theta =$	$\sin \theta \$	${\sqrt {1-\cos ^{2}\theta }}$	${\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}$	${\frac {1}{\sqrt {1+\cot ^{2}\theta }}}$	${\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}$	${\frac {1}{\csc \theta }}$
$\cos \theta =$	${\sqrt {1-\sin ^{2}\theta }}$	$\cos \theta \$	${\frac {1}{\sqrt {1+\tan ^{2}\theta }}}$	${\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}$	${\frac {1}{\sec \theta }}$	${\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}$
$\tan \theta =$	${\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}$	${\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}$	$\tan \theta \$	${\frac {1}{\cot \theta }}$	${\sqrt {\sec ^{2}\theta -1}}$	${\frac {1}{\sqrt {\csc ^{2}\theta -1}}}$
$\cot \theta =$	${{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }$	${\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}$	${1 \over \tan \theta }$	$\cot \theta \$	${1 \over {\sqrt {\sec ^{2}\theta -1}}}$	${\sqrt {\csc ^{2}\theta -1}}$
$\sec \theta =$	${1 \over {\sqrt {1-\sin ^{2}\theta }}}$	${1 \over \cos \theta }$	${\sqrt {1+\tan ^{2}\theta }}$	${{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }$	$\sec \theta \$	${\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}$
$\csc \theta =$	${1 \over \sin \theta }$	${1 \over {\sqrt {1-\cos ^{2}\theta }}}$	${{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }$	${\sqrt {1+\cot ^{2}\theta }}$	${\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}$	$\csc \theta \$

各種角度對應到0～45度的三角函數

角度	0～45度	45～90度	90～135度	135～180度	180～225度	225～270度	270～315度	315～360度
弧度	${\frac {0}{4}}\leq x\leq {\frac {\pi }{4}}\$	${\frac {\pi }{4}}\leq x\leq {\frac {2\pi }{4}}\$	${\frac {2\pi }{4}}\leq x\leq {\frac {3\pi }{4}}\$	${\frac {3\pi }{4}}\leq x\leq {\frac {4\pi }{4}}\$	${\frac {4\pi }{4}}\leq x\leq {\frac {5\pi }{4}}\$	${\frac {5\pi }{4}}\leq x\leq {\frac {6\pi }{4}}\$	${\frac {6\pi }{4}}\leq x\leq {\frac {7\pi }{4}}\$	${\frac {7\pi }{4}}\leq x\leq {\frac {8\pi }{4}}\$
$\theta \$	$\theta \ =x$	$\theta \ ={\frac {\pi }{2}}-x\$	$\theta \ =x-{\frac {\pi }{2}}\$	$\theta \ =\pi -x\$	$\theta \ =x-\pi \$	$\theta \ ={\frac {3\pi }{2}}-x\$	$\theta \ =x-{\frac {3\pi }{2}}$	$\theta \ =2\pi -x\$
$\sin x=\$	$\sin \theta \$	$+\cos \theta \$	$+\cos \theta \$	$+\sin \theta \$	$-\sin \theta \$	$-\cos \theta \$	$-\cos \theta \$	$-\sin \theta \$
$\cos x=\$	$\cos \theta \$	$+\sin \theta \$	$-\sin \theta \$	$-\cos \theta \$	$-\cos \theta \$	$-\sin \theta \$	$+\sin \theta \$	$+\cos \theta \$
$\tan x=\$	$\tan \theta \$	$+\cot \theta \$	$-\cot \theta \$	$-\tan \theta \$	$+\tan \theta \$	$+\cot \theta \$	$-\cot \theta \$	$-\tan \theta \$
$\cot x=\$	$\cot \theta \$	$+\tan \theta \$	$-\tan \theta \$	$-\cot \theta \$	$+\cot \theta \$	$+\tan \theta \$	$-\tan \theta \$	$-\cot \theta \$
$\sec x=\$	$\sec \theta \$	$+\csc \theta \$	$-\csc \theta \$	$-\sec \theta \$	$-\sec \theta \$	$-\csc \theta \$	$+\csc \theta \$	$+\sec \theta \$
$\csc x=\$	$\csc \theta \$	$+\sec \theta \$	$+\sec \theta \$	$+\csc \theta \$	$-\csc \theta \$	$-\sec \theta \$	$-\sec \theta \$	$-\csc \theta \$