| 關鍵字 |
說明 |
|
| 常用的特別角 |
| |
\(\sin \theta \) |
\(\cos \theta \) |
\(\tan \theta \) |
| \(\theta = {30^ \circ }\) |
\(\frac{1}{2}\) |
\(\frac{{\sqrt 3 }}{2}\) |
\(\frac{1}{{\sqrt 3 }}\) |
| \(\theta = {45^ \circ }\) |
\(\frac{{\sqrt 2 }}{2}\) |
\(\frac{{\sqrt 2 }}{2}\) |
\(1\) |
| \(\theta = {60^ \circ }\) |
\(\frac{{\sqrt 3 }}{2}\) |
\(\frac{1}{2}\) |
\(\sqrt 3 \) |
|
| 廣義三角函數求法 |
方法1
以 \(x\) 軸( \({180^ \circ }\) )為基準，先判斷特別角，再補正負號
方法2
遇到大角度時以 \(x\) 軸( \({360^ \circ }\) )為除數，除後之餘數在應用方法1 |
| 負角關係 |
\(\cos ( - \theta ) = \cos \theta \) ， \(\sin ( - \theta ) = - \sin \theta |
| 互補關係 |
\(\cos ({180^ \circ } - \theta ) = - \cos \theta \) ， \(\sin ({180^ \circ } - \theta ) = \sin \theta |
| 弧長 |
半徑為 \(r\) 的圓，
圓心角為 \(\theta \)弧長 \(L = \frac{\theta }{{{{360}^ \circ }}}2\pi r = \frac{\theta }{{{{180}^ \circ }}}\pi r = r\theta \) |
| 扇形面積 |
半徑為 \(r\) 的圓，
圓心角為 \(\theta \)扇形面積 \(A = \frac{\theta }{{{{360}^ \circ }}}\pi {r^2} = \frac{\theta }{2}{r^2} = \frac{1}{2}{r^2}\theta \) |
| 平方和關係 |
\({\sin ^2}\theta + {\cos ^2}\theta = {1^2}\) ， \({\tan ^2}\theta + {1^2} = {\sec ^2}\theta \) ， \({1^2} + {\cot ^2}\theta = {\csc ^2}\theta \) |
| 商數關係 |
\(\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}\) ， \(\cot \theta = \frac{{\cos \theta }}{{\sin \theta }}\) |
| 倒數關係 |
\(\left( {\sin \theta } \right)\left( {\csc \theta } \right) = 1\) ， \(\left( {\cos \theta } \right)\left( {\sec \theta } \right) = 1\) ， \(\left( {\tan \theta } \right)\left( {\cot \theta } \right) = 1\) |
| 正餘弦函數的週期 |
(1) \(y = \sin x\) 之(最小)週期為 \(2\pi \)
(2) \(y = \cos x\) 之(最小)週期為 \(2\pi \)
|
| 正餘切弦函數的週期 |
(1) \(y = \tan x\) 之(最小)週為為 \(\pi \)
(2) \(y = \cot x\) 之(最小)週期為 \(\pi \) |
| 正弦定律 |
\(\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}} = 2R\) |
| 餘弦定律 |
設 \(\vartriangle ABC\) 中， \(\angle A\) 之對邊為 \(a\) ， \(\angle B\) 之對邊為 \(b\) ， \(\angle C\) 之對邊為 \(c\)，則
\({a^2} = {b^2} + {c^2} - 2bc\cos A\)
\({b^2} = {a^2} + {c^2} - 2ac\cos B\)
\({c^2} = {a^2} + {b^2} - 2ab\cos C\) |
| 海倫公式 |
若已知 \(\vartriangle ABC\) 中三邊長各為 \(a\) , \(b\) , \(c\) , \(s\) 表示周長之半，即 \(s = \frac{{a + b + c}}{2}\) ，
則 \(\vartriangle ABC\) 之面積為 \(\sqrt {s\;(s - a)\,\,(s - b)\,\,(s - c)} \) |
