三倍角公式
sin(3α) = 3sinα-4sin^3α = 4sinα·sin(π/3+α)sin(π/3-α) cos(3α) = 4cos^3α-3cos α = 4cosα·cos(π/3+α)cos(π/3-α) tan(3α) = (3tanα-tan^3α)/(1-3tan^2α) =
tan αtan(π/3+α)tan(π/3-α)
2推导过程
1.sin3a
=sin(2a+a)
=sin2acosa+cos2asina
=2sina(1-sin²a)+(1-2sin²a)sina
=3sina-4sin³a
2.cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos²a-1)cosa-2(1-cos²a)cosa
=4cos³a-3cosa
（1）sin3a=3sina-4sin³a
=4sina(3/4-sin²a)
=4sina[(√3/2)²-sin²a]
=4sina(sin²60°-sin²a)
=4sina(sin60°+sina)(sin60°-sina)
=4sina*2sin[(60°+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°+a)/2]
=4sinasin(60°+a)sin(60°-a)
（2）cos3a=4cos³a-3cosa
=4cosa(cos²a-3/4)
=4cosa[cos²a-(√3/2)²]
=4cosa(cos²a-cos²30°)
=4cosa(cosa+cos30°)(cosa-cos30°)
=4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)/2]}
=-4cosasin(a+30°)sin(a-30°)
=-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)]
=-4cosacos(60°-a)[-cos(60°+a)]
=4cosacos(60°-a)cos(60°+a)
综上述两式相比可得
tan3a=tanatan(60°-a)tan(60°+a)
3联想记忆
. 记忆方法：谐音、联想
正弦三倍角：3元 减 4元3角（欠债了(被减成负数) ，所以要“挣钱”(音似“正弦”)）
余弦三倍角：4元3角 减 3元（减完之后还有“余”）
☆☆注意函数名，即正弦的三倍角都用正弦表示，余弦的三倍角都用余弦表示。 ······································································································································ 另一个记忆方法：1. 正弦三倍角 ：3 1 4 3 （3sina-4sin^3a）
2. 余弦三倍角：4 3 3 1 （4cos^3a-3cosa）
注意 ①正弦里函数名都为sin 余弦里函数名都为cos
②中间都为减号
三倍角公式
sin(3α) = 3sinα-4sin^3α = 4sinα·sin(π/3+α)sin(π/3-α) cos(3α) = 4cos^3α-3cos α = 4cosα·cos(π/3+α)cos(π/3-α) tan(3α) = (3tanα-tan^3α)/(1-3tan^2α) =
tan αtan(π/3+α)tan(π/3-α)
2推导过程
1.sin3a
=sin(2a+a)
=sin2acosa+cos2asina
=2sina(1-sin²a)+(1-2sin²a)sina
=3sina-4sin³a
2.cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos²a-1)cosa-2(1-cos²a)cosa
=4cos³a-3cosa
（1）sin3a=3sina-4sin³a
=4sina(3/4-sin²a)
=4sina[(√3/2)²-sin²a]
=4sina(sin²60°-sin²a)
=4sina(sin60°+sina)(sin60°-sina)
=4sina*2sin[(60°+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°+a)/2]
=4sinasin(60°+a)sin(60°-a)
（2）cos3a=4cos³a-3cosa
=4cosa(cos²a-3/4)
=4cosa[cos²a-(√3/2)²]
=4cosa(cos²a-cos²30°)
=4cosa(cosa+cos30°)(cosa-cos30°)
=4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)/2]}
=-4cosasin(a+30°)sin(a-30°)
=-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)]
=-4cosacos(60°-a)[-cos(60°+a)]
=4cosacos(60°-a)cos(60°+a)
综上述两式相比可得
tan3a=tanatan(60°-a)tan(60°+a)
3联想记忆
. 记忆方法：谐音、联想
正弦三倍角：3元 减 4元3角（欠债了(被减成负数) ，所以要“挣钱”(音似“正弦”)）
余弦三倍角：4元3角 减 3元（减完之后还有“余”）
☆☆注意函数名，即正弦的三倍角都用正弦表示，余弦的三倍角都用余弦表示。 ······································································································································ 另一个记忆方法：1. 正弦三倍角 ：3 1 4 3 （3sina-4sin^3a）
2. 余弦三倍角：4 3 3 1 （4cos^3a-3cosa）
注意 ①正弦里函数名都为sin 余弦里函数名都为cos
②中间都为减号