三角函数

三角函数基础知识

一、定义:

正弦: (sin A = frac{a}{c} = frac{对边}{斜边})

余弦: (cos A = frac{b}{c} = frac{邻边}{斜边})

正切: ( an A = frac{a}{b} = frac{对边}{邻边})

余切: (cot A = frac{b}{a} = frac{邻边}{对边})

特殊性质: |(sin alpha)| (leq 1) , |(cos alpha)| (leq 1)

二、特殊角三角函数

	(0°)	(15°)	(30°)	(45°)	(60°)	(75°)	(90°)
(sin)	(0)	(frac{sqrt{6}-sqrt{2}}{4})	(frac{1}{2})	(frac{sqrt{2}}{2})	(frac{sqrt{3}}{2})	(frac{sqrt{6}+sqrt{2}}{4})	(1)
(cos)	(0)	(frac{sqrt{6}+sqrt{2}}{4})	(frac{sqrt{3}}{2})	(frac{sqrt{2}}{2})	(frac{1}{2})	(frac{sqrt{6}-sqrt{2}}{4})	(0)
( an)	(0)	(2-sqrt{3})	(frac{sqrt{3}}{3})	(1)	(sqrt{3})	(2+sqrt{3})	/
(cot)	/	(2+sqrt{3})	(sqrt{3})	(1)	(frac{sqrt{3}}{3})	(2-sqrt{3})	(0)

三、基本公式

1. 若 (angle A + angle B = 90°) ,则 (sin A = cos B) , ( an A = cot B)

2. ( an A cdot cot A = 1)

3. ( an A = frac{sin A}{cos A})

4. (sin^2 A + cos^2 A = 1)

四、三角形面积公式

五、两角和差公式

(sin (alpha pm eta) = sin alpha cdot cos eta pm sin eta cdot cos alpha)

(cos (alpha pm eta) = cos alpha cdot cos eta mp sin alpha cdot sin eta)

( an (alpha pm eta) = frac{ an alpha pm an eta}{1 mp an alpha cdot an eta})

六、倍角公式

(sin 2 alpha = 2 sin alpha cdot cos alpha)

(cos 2 alpha = cos^2 alpha - sin^2 alpha = 1-2sin^2 alpha = 2cos^2 alpha -1)

( an 2 alpha = frac{2 an alpha}{1- an^2 alpha})

七、直线斜率

1. |(k)| (= an heta) , ( heta) 为该直线与 (x) 轴相交所形成的最小夹角