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616170155:三角函数及解三角形公式(精简版)

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三角函数及解三角形公式大全(精简版)工作室微信:SQ-math 1.诱导公式:

?
2

? k ? ? ? ? ( k ? Z )记忆公式口诀:奇变偶不变、符号看象限
2 2

2.同角的三角函数关系: sin x ? cos x ? 1 ; tan x ? 3.两角和与差的三角函数:

sin x ; tan x ? cot x ? 1 cos x

sin( x ? y) ? sin x cos y ? cos x sin y ; cos( x ? y) ? cos x cos y ? sin x sin y ; tan( x ? y) ? tan x ? tan y
4.辅助角公式: a sin ? x ? b cos ? x ? a 2 ? b2 (sin ? x ?

1 ? tan x tan y

a a ?b
2 2

? cos ? x ?

b a ? b2
2

)

? a 2 ? b 2 (sin ? x ? cos ? ? cos ? x ? sin ? ) ? a 2 ? b 2 sin(? x ? ? )

2 tan x 2 2 2 2 ;cos 2 x ? cos x ? sin x ? 1 ? 2sin x ? 2cos x ? 1 2 1 ? tan x 1 1 1 2 2 6.降幂公式: sin x cos x ? sin 2 x ; sin x ? (1 ? cos 2 x) ; cos x ? (1 ? cos 2 x ) 2 2 2 a b c ? ? ? 2 R ( R 为 ?ABC 的外接圆半径) 7.正弦定理: “对边对角关系” sin A sin B sin C b sin A asinB 变式 1: a ? ;变式 2: sin A ? ;变式 3: a ? 2 R sin A ; b ? 2 R sin B ; c ? 2 R sin C ; sin B b a b c 变式 4: sin A ? ; sin B ? ; sin C ? ;变式 5: a : b : c ? sin A : sin B : sin C 2R 2R 2R
5.二倍角公式:sin 2 x ? 2sin x cos x ;tan 2 x ? 8.余弦定理: a ? b ? c ? 2bc cos A ; b ? a ? c ? 2ac cos B ; c ? a ? b ? 2ab cos C “三边一角关系”
2 2 2 2 2 2 2 2 2

b2 ? c2 ? a 2 a 2 ? c2 ? b2 a 2 ? b2 ? c 2 变式 1: cos A ? ; cos B ? ; cos C ? 2ab 2bc 2ac
变式 2: b ? c ? a ? 2bc cos A ; a ? c ? b ? 2ac cos B ; a ? b ? c ? 2ab cos C
2 2 2 2 2 2 2 2 2

1 1 1 1 ah ; (2) S ? ab sin C ? bc sin A ? ac sin B ; 2 2 2 2 1 abc ? 2 R 2 sin A sin B sin C ( R 为 ?ABC 的外接圆半径) (3) S ? ; (4) S ? (a ? b ? c )r ( r 为 ?ABC 的内 2 4R 1 切圆半径) ; (5)海伦公式: S ? p( p ? a)( p ? b)( p ? c) ,其中 p ? ( a ? b ? c ) . 2 A ? B?C 10.三角形的常用结论: (1) A ? B ? C ? ? ; ? ? ; 2 2 2
9.三角形面积公式: (1) S ? (2) sin A ? sin( B ? C ) ; sin B ? sin( A ? C ) ; sin C ? sin( A ? B) ; (3) cos A ? ? cos( B ? C ) ; cos B ? ? cos( A ? C ) ; cos C ? ? cos( A ? B) ; (4) tan A ? ? tan( B ? C ) ; tan B ? ? tan( A ? C ) ; tan C ? ? tan( A ? B) ; (5) sin

A B?C B A?C C A? B ? cos( ) ; sin ? cos( ) ; sin ? cos( ); 2 2 2 2 2 2

(6)在锐角 ?ABC 中,无论 A, B 为何角,一定有 sin A ? cos B ; (7)在 ?ABC 中, “ a ? b ” 等价于“ A ? B ” 等价于“ sin A ? sin B ”.


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