使用者:P1ayer/備頁/三角函數轉換表維基百科,自由的 encyclopedia 當平面上的三點O、C、A的連線, O C ¯ {\displaystyle {\overline {OC}}} 、 C A ¯ {\displaystyle {\overline {CA}}} 、 O A ¯ {\displaystyle {\overline {OA}}} ,構成一個直角三角形。對於 O C ¯ {\displaystyle {\overline {OC}}} 與 O A ¯ {\displaystyle {\overline {OA}}} 的夾角 ∠ θ {\displaystyle {\angle \theta }} 而言: 對邊(opposite) y = C A ¯ {\displaystyle y={\overline {CA}}} 鄰邊(adjacent) x = O C ¯ {\displaystyle x={\overline {OC}}} 斜邊(hypotenuse) r = O A ¯ {\displaystyle r={\overline {OA}}} More information , ... 函數 英語 簡寫 定義 關係 正弦 Sine sin y r {\displaystyle {\frac {y}{r}}} sin θ = 1 csc θ {\displaystyle \sin \theta ={\frac {1}{\csc \theta }}} 餘弦 Cosine cos x r {\displaystyle {\frac {x}{r}}} cos θ = 1 sec θ {\displaystyle \cos \theta ={\frac {1}{\sec \theta }}\,} 正切 Tangent tan y x {\displaystyle {\frac {y}{x}}} tan θ = sin θ cos θ = 1 cot θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {1}{\cot \theta }}} 餘切 Cotangent cot x y {\displaystyle {\frac {x}{y}}} cot θ = cos θ sin θ = 1 tan θ {\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}={\frac {1}{\tan \theta }}} 正割 Secant sec r x {\displaystyle {\frac {r}{x}}} sec θ = 1 cos θ {\displaystyle \sec \theta ={\frac {1}{\cos \theta }}} 餘割 Cosecant csc r y {\displaystyle {\frac {r}{y}}} csc θ = 1 sin θ {\displaystyle \csc \theta ={\frac {1}{\sin \theta }}} Close 更多資訊:三角恆等式 § 基本關係 基本關係 More information , ... 函數 sin cos tan cot sec csc sin θ = {\displaystyle \sin \theta =} sin θ {\displaystyle \sin \theta \ } 1 − cos 2 θ {\displaystyle {\sqrt {1-\cos ^{2}\theta }}} tan θ 1 + tan 2 θ {\displaystyle {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}} 1 1 + cot 2 θ {\displaystyle {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}} sec 2 θ − 1 sec θ {\displaystyle {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}} 1 csc θ {\displaystyle {\frac {1}{\csc \theta }}} cos θ = {\displaystyle \cos \theta =} 1 − sin 2 θ {\displaystyle {\sqrt {1-\sin ^{2}\theta }}} cos θ {\displaystyle \cos \theta \ } 1 1 + tan 2 θ {\displaystyle {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}} cot θ 1 + cot 2 θ {\displaystyle {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}} 1 sec θ {\displaystyle {\frac {1}{\sec \theta }}} csc 2 θ − 1 csc θ {\displaystyle {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}} tan θ = {\displaystyle \tan \theta =} sin θ 1 − sin 2 θ {\displaystyle {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}} 1 − cos 2 θ cos θ {\displaystyle {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}} tan θ {\displaystyle \tan \theta \ } 1 cot θ {\displaystyle {\frac {1}{\cot \theta }}} sec 2 θ − 1 {\displaystyle {\sqrt {\sec ^{2}\theta -1}}} 1 csc 2 θ − 1 {\displaystyle {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}} cot θ = {\displaystyle \cot \theta =} 1 − sin 2 θ sin θ {\displaystyle {{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }} cos θ 1 − cos 2 θ {\displaystyle {\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}} 1 tan θ {\displaystyle {1 \over \tan \theta }} cot θ {\displaystyle \cot \theta \ } 1 sec 2 θ − 1 {\displaystyle {1 \over {\sqrt {\sec ^{2}\theta -1}}}} csc 2 θ − 1 {\displaystyle {\sqrt {\csc ^{2}\theta -1}}} sec θ = {\displaystyle \sec \theta =} 1 1 − sin 2 θ {\displaystyle {1 \over {\sqrt {1-\sin ^{2}\theta }}}} 1 cos θ {\displaystyle {1 \over \cos \theta }} 1 + tan 2 θ {\displaystyle {\sqrt {1+\tan ^{2}\theta }}} 1 + cot 2 θ cot θ {\displaystyle {{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }} sec θ {\displaystyle \sec \theta \ } csc θ csc 2 θ − 1 {\displaystyle {\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}} csc θ = {\displaystyle \csc \theta =} 1 sin θ {\displaystyle {1 \over \sin \theta }} 1 1 − cos 2 θ {\displaystyle {1 \over {\sqrt {1-\cos ^{2}\theta }}}} 1 + tan 2 θ tan θ {\displaystyle {{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }} 1 + cot 2 θ {\displaystyle {\sqrt {1+\cot ^{2}\theta }}} sec θ sec 2 θ − 1 {\displaystyle {\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}} csc θ {\displaystyle \csc \theta \ } Close 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 0 4 ≤ x ≤ π 4 {\displaystyle {\frac {0}{4}}\leq x\leq {\frac {\pi }{4}}\ } π 4 ≤ x ≤ 2 π 4 {\displaystyle {\frac {\pi }{4}}\leq x\leq {\frac {2\pi }{4}}\ } 2 π 4 ≤ x ≤ 3 π 4 {\displaystyle {\frac {2\pi }{4}}\leq x\leq {\frac {3\pi }{4}}\ } 3 π 4 ≤ x ≤ 4 π 4 {\displaystyle {\frac {3\pi }{4}}\leq x\leq {\frac {4\pi }{4}}\ } 4 π 4 ≤ x ≤ 5 π 4 {\displaystyle {\frac {4\pi }{4}}\leq x\leq {\frac {5\pi }{4}}\ } 5 π 4 ≤ x ≤ 6 π 4 {\displaystyle {\frac {5\pi }{4}}\leq x\leq {\frac {6\pi }{4}}\ } 6 π 4 ≤ x ≤ 7 π 4 {\displaystyle {\frac {6\pi }{4}}\leq x\leq {\frac {7\pi }{4}}\ } 7 π 4 ≤ x ≤ 8 π 4 {\displaystyle {\frac {7\pi }{4}}\leq x\leq {\frac {8\pi }{4}}\ } θ {\displaystyle \theta \ } θ = x {\displaystyle \theta \ =x} θ = π 2 − x {\displaystyle \theta \ ={\frac {\pi }{2}}-x\ } θ = x − π 2 {\displaystyle \theta \ =x-{\frac {\pi }{2}}\ } θ = π − x {\displaystyle \theta \ =\pi -x\ } θ = x − π {\displaystyle \theta \ =x-\pi \ } θ = 3 π 2 − x {\displaystyle \theta \ ={\frac {3\pi }{2}}-x\ } θ = x − 3 π 2 {\displaystyle \theta \ =x-{\frac {3\pi }{2}}} θ = 2 π − x {\displaystyle \theta \ =2\pi -x\ } sin x = {\displaystyle \sin x=\ } sin θ {\displaystyle \sin \theta \ } + cos θ {\displaystyle +\cos \theta \ } + cos θ {\displaystyle +\cos \theta \ } + sin θ {\displaystyle +\sin \theta \ } − sin θ {\displaystyle -\sin \theta \ } − cos θ {\displaystyle -\cos \theta \ } − cos θ {\displaystyle -\cos \theta \ } − sin θ {\displaystyle -\sin \theta \ } cos x = {\displaystyle \cos x=\ } cos θ {\displaystyle \cos \theta \ } + sin θ {\displaystyle +\sin \theta \ } − sin θ {\displaystyle -\sin \theta \ } − cos θ {\displaystyle -\cos \theta \ } − cos θ {\displaystyle -\cos \theta \ } − sin θ {\displaystyle -\sin \theta \ } + sin θ {\displaystyle +\sin \theta \ } + cos θ {\displaystyle +\cos \theta \ } tan x = {\displaystyle \tan x=\ } tan θ {\displaystyle \tan \theta \ } + cot θ {\displaystyle +\cot \theta \ } − cot θ {\displaystyle -\cot \theta \ } − tan θ {\displaystyle -\tan \theta \ } + tan θ {\displaystyle +\tan \theta \ } + cot θ {\displaystyle +\cot \theta \ } − cot θ {\displaystyle -\cot \theta \ } − tan θ {\displaystyle -\tan \theta \ } cot x = {\displaystyle \cot x=\ } cot θ {\displaystyle \cot \theta \ } + tan θ {\displaystyle +\tan \theta \ } − tan θ {\displaystyle -\tan \theta \ } − cot θ {\displaystyle -\cot \theta \ } + cot θ {\displaystyle +\cot \theta \ } + tan θ {\displaystyle +\tan \theta \ } − tan θ {\displaystyle -\tan \theta \ } − cot θ {\displaystyle -\cot \theta \ } sec x = {\displaystyle \sec x=\ } sec θ {\displaystyle \sec \theta \ } + csc θ {\displaystyle +\csc \theta \ } − csc θ {\displaystyle -\csc \theta \ } − sec θ {\displaystyle -\sec \theta \ } − sec θ {\displaystyle -\sec \theta \ } − csc θ {\displaystyle -\csc \theta \ } + csc θ {\displaystyle +\csc \theta \ } + sec θ {\displaystyle +\sec \theta \ } csc x = {\displaystyle \csc x=\ } csc θ {\displaystyle \csc \theta \ } + sec θ {\displaystyle +\sec \theta \ } + sec θ {\displaystyle +\sec \theta \ } + csc θ {\displaystyle +\csc \theta \ } − csc θ {\displaystyle -\csc \theta \ } − sec θ {\displaystyle -\sec \theta \ } − sec θ {\displaystyle -\sec \theta \ } − csc θ {\displaystyle -\csc \theta \ } 基本函數
當平面上的三點O、C、A的連線, O C ¯ {\displaystyle {\overline {OC}}} 、 C A ¯ {\displaystyle {\overline {CA}}} 、 O A ¯ {\displaystyle {\overline {OA}}} ,構成一個直角三角形。對於 O C ¯ {\displaystyle {\overline {OC}}} 與 O A ¯ {\displaystyle {\overline {OA}}} 的夾角 ∠ θ {\displaystyle {\angle \theta }} 而言: 對邊(opposite) y = C A ¯ {\displaystyle y={\overline {CA}}} 鄰邊(adjacent) x = O C ¯ {\displaystyle x={\overline {OC}}} 斜邊(hypotenuse) r = O A ¯ {\displaystyle r={\overline {OA}}} More information , ... 函數 英語 簡寫 定義 關係 正弦 Sine sin y r {\displaystyle {\frac {y}{r}}} sin θ = 1 csc θ {\displaystyle \sin \theta ={\frac {1}{\csc \theta }}} 餘弦 Cosine cos x r {\displaystyle {\frac {x}{r}}} cos θ = 1 sec θ {\displaystyle \cos \theta ={\frac {1}{\sec \theta }}\,} 正切 Tangent tan y x {\displaystyle {\frac {y}{x}}} tan θ = sin θ cos θ = 1 cot θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {1}{\cot \theta }}} 餘切 Cotangent cot x y {\displaystyle {\frac {x}{y}}} cot θ = cos θ sin θ = 1 tan θ {\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}={\frac {1}{\tan \theta }}} 正割 Secant sec r x {\displaystyle {\frac {r}{x}}} sec θ = 1 cos θ {\displaystyle \sec \theta ={\frac {1}{\cos \theta }}} 餘割 Cosecant csc r y {\displaystyle {\frac {r}{y}}} csc θ = 1 sin θ {\displaystyle \csc \theta ={\frac {1}{\sin \theta }}} Close 更多資訊:三角恆等式 § 基本關係 基本關係 More information , ... 函數 sin cos tan cot sec csc sin θ = {\displaystyle \sin \theta =} sin θ {\displaystyle \sin \theta \ } 1 − cos 2 θ {\displaystyle {\sqrt {1-\cos ^{2}\theta }}} tan θ 1 + tan 2 θ {\displaystyle {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}} 1 1 + cot 2 θ {\displaystyle {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}} sec 2 θ − 1 sec θ {\displaystyle {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}} 1 csc θ {\displaystyle {\frac {1}{\csc \theta }}} cos θ = {\displaystyle \cos \theta =} 1 − sin 2 θ {\displaystyle {\sqrt {1-\sin ^{2}\theta }}} cos θ {\displaystyle \cos \theta \ } 1 1 + tan 2 θ {\displaystyle {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}} cot θ 1 + cot 2 θ {\displaystyle {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}} 1 sec θ {\displaystyle {\frac {1}{\sec \theta }}} csc 2 θ − 1 csc θ {\displaystyle {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}} tan θ = {\displaystyle \tan \theta =} sin θ 1 − sin 2 θ {\displaystyle {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}} 1 − cos 2 θ cos θ {\displaystyle {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}} tan θ {\displaystyle \tan \theta \ } 1 cot θ {\displaystyle {\frac {1}{\cot \theta }}} sec 2 θ − 1 {\displaystyle {\sqrt {\sec ^{2}\theta -1}}} 1 csc 2 θ − 1 {\displaystyle {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}} cot θ = {\displaystyle \cot \theta =} 1 − sin 2 θ sin θ {\displaystyle {{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }} cos θ 1 − cos 2 θ {\displaystyle {\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}} 1 tan θ {\displaystyle {1 \over \tan \theta }} cot θ {\displaystyle \cot \theta \ } 1 sec 2 θ − 1 {\displaystyle {1 \over {\sqrt {\sec ^{2}\theta -1}}}} csc 2 θ − 1 {\displaystyle {\sqrt {\csc ^{2}\theta -1}}} sec θ = {\displaystyle \sec \theta =} 1 1 − sin 2 θ {\displaystyle {1 \over {\sqrt {1-\sin ^{2}\theta }}}} 1 cos θ {\displaystyle {1 \over \cos \theta }} 1 + tan 2 θ {\displaystyle {\sqrt {1+\tan ^{2}\theta }}} 1 + cot 2 θ cot θ {\displaystyle {{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }} sec θ {\displaystyle \sec \theta \ } csc θ csc 2 θ − 1 {\displaystyle {\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}} csc θ = {\displaystyle \csc \theta =} 1 sin θ {\displaystyle {1 \over \sin \theta }} 1 1 − cos 2 θ {\displaystyle {1 \over {\sqrt {1-\cos ^{2}\theta }}}} 1 + tan 2 θ tan θ {\displaystyle {{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }} 1 + cot 2 θ {\displaystyle {\sqrt {1+\cot ^{2}\theta }}} sec θ sec 2 θ − 1 {\displaystyle {\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}} csc θ {\displaystyle \csc \theta \ } Close 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 0 4 ≤ x ≤ π 4 {\displaystyle {\frac {0}{4}}\leq x\leq {\frac {\pi }{4}}\ } π 4 ≤ x ≤ 2 π 4 {\displaystyle {\frac {\pi }{4}}\leq x\leq {\frac {2\pi }{4}}\ } 2 π 4 ≤ x ≤ 3 π 4 {\displaystyle {\frac {2\pi }{4}}\leq x\leq {\frac {3\pi }{4}}\ } 3 π 4 ≤ x ≤ 4 π 4 {\displaystyle {\frac {3\pi }{4}}\leq x\leq {\frac {4\pi }{4}}\ } 4 π 4 ≤ x ≤ 5 π 4 {\displaystyle {\frac {4\pi }{4}}\leq x\leq {\frac {5\pi }{4}}\ } 5 π 4 ≤ x ≤ 6 π 4 {\displaystyle {\frac {5\pi }{4}}\leq x\leq {\frac {6\pi }{4}}\ } 6 π 4 ≤ x ≤ 7 π 4 {\displaystyle {\frac {6\pi }{4}}\leq x\leq {\frac {7\pi }{4}}\ } 7 π 4 ≤ x ≤ 8 π 4 {\displaystyle {\frac {7\pi }{4}}\leq x\leq {\frac {8\pi }{4}}\ } θ {\displaystyle \theta \ } θ = x {\displaystyle \theta \ =x} θ = π 2 − x {\displaystyle \theta \ ={\frac {\pi }{2}}-x\ } θ = x − π 2 {\displaystyle \theta \ =x-{\frac {\pi }{2}}\ } θ = π − x {\displaystyle \theta \ =\pi -x\ } θ = x − π {\displaystyle \theta \ =x-\pi \ } θ = 3 π 2 − x {\displaystyle \theta \ ={\frac {3\pi }{2}}-x\ } θ = x − 3 π 2 {\displaystyle \theta \ =x-{\frac {3\pi }{2}}} θ = 2 π − x {\displaystyle \theta \ =2\pi -x\ } sin x = {\displaystyle \sin x=\ } sin θ {\displaystyle \sin \theta \ } + cos θ {\displaystyle +\cos \theta \ } + cos θ {\displaystyle +\cos \theta \ } + sin θ {\displaystyle +\sin \theta \ } − sin θ {\displaystyle -\sin \theta \ } − cos θ {\displaystyle -\cos \theta \ } − cos θ {\displaystyle -\cos \theta \ } − sin θ {\displaystyle -\sin \theta \ } cos x = {\displaystyle \cos x=\ } cos θ {\displaystyle \cos \theta \ } + sin θ {\displaystyle +\sin \theta \ } − sin θ {\displaystyle -\sin \theta \ } − cos θ {\displaystyle -\cos \theta \ } − cos θ {\displaystyle -\cos \theta \ } − sin θ {\displaystyle -\sin \theta \ } + sin θ {\displaystyle +\sin \theta \ } + cos θ {\displaystyle +\cos \theta \ } tan x = {\displaystyle \tan x=\ } tan θ {\displaystyle \tan \theta \ } + cot θ {\displaystyle +\cot \theta \ } − cot θ {\displaystyle -\cot \theta \ } − tan θ {\displaystyle -\tan \theta \ } + tan θ {\displaystyle +\tan \theta \ } + cot θ {\displaystyle +\cot \theta \ } − cot θ {\displaystyle -\cot \theta \ } − tan θ {\displaystyle -\tan \theta \ } cot x = {\displaystyle \cot x=\ } cot θ {\displaystyle \cot \theta \ } + tan θ {\displaystyle +\tan \theta \ } − tan θ {\displaystyle -\tan \theta \ } − cot θ {\displaystyle -\cot \theta \ } + cot θ {\displaystyle +\cot \theta \ } + tan θ {\displaystyle +\tan \theta \ } − tan θ {\displaystyle -\tan \theta \ } − cot θ {\displaystyle -\cot \theta \ } sec x = {\displaystyle \sec x=\ } sec θ {\displaystyle \sec \theta \ } + csc θ {\displaystyle +\csc \theta \ } − csc θ {\displaystyle -\csc \theta \ } − sec θ {\displaystyle -\sec \theta \ } − sec θ {\displaystyle -\sec \theta \ } − csc θ {\displaystyle -\csc \theta \ } + csc θ {\displaystyle +\csc \theta \ } + sec θ {\displaystyle +\sec \theta \ } csc x = {\displaystyle \csc x=\ } csc θ {\displaystyle \csc \theta \ } + sec θ {\displaystyle +\sec \theta \ } + sec θ {\displaystyle +\sec \theta \ } + csc θ {\displaystyle +\csc \theta \ } − csc θ {\displaystyle -\csc \theta \ } − sec θ {\displaystyle -\sec \theta \ } − sec θ {\displaystyle -\sec \theta \ } − csc θ {\displaystyle -\csc \theta \ } 基本函數