三角函数 积化和差、和差化积公式
积化和差公式
公式1
s i n A ⋅ s i n B = − 1 2 [ c o s ( A + B ) − c o s ( A − B ) ] \mathrm{sin}A\cdot\mathrm{sin}B=-\dfrac{1}{2}[\mathrm{cos}(A+B)-\mathrm{cos}(A-B)] sinA⋅sinB=−21[cos(A+B)−cos(A−B)].
− 1 2 [ c o s ( A + B ) − c o s ( A − B ) ] -\dfrac{1}{2}[\mathrm{cos}(A+B)-\mathrm{cos}(A-B)] −21[cos(A+B)−cos(A−B)]
= − 1 2 ( c o s A ⋅ c o s B − s i n A ⋅ s i n B − c o s A ⋅ c o s B − s i n A ⋅ s i n B ) =-\dfrac{1}{2}(\mathrm{cos}A\cdot\mathrm{cos}B-\mathrm{sin}A\cdot\mathrm{sin}B-\mathrm{cos}A\cdot\mathrm{cos}B-\mathrm{sin}A\cdot\mathrm{sin}B) =−21(cosA⋅cosB−sinA⋅sinB−cosA⋅cosB−sinA⋅sinB)
= s i n A ⋅ s i n B =\mathrm{sin}A\cdot\mathrm{sin}B =sinA⋅sinB
公式2
c o s A ⋅ c o s B = 1 2 [ c o s ( A + B ) + c o s ( A − B ) ] \mathrm{cos}A\cdot\mathrm{cosB}=\dfrac{1}{2}[\mathrm{cos}(A+B)+\mathrm{cos}(A-B)] cosA⋅cosB=21[cos(A+B)+cos(A−B)].
1 2 [ c o s ( A + B ) + c o s ( A − B ) ] \dfrac{1}{2}[\mathrm{cos}(A+B)+\mathrm{cos}(A-B)] 21[cos(A+B)+cos(A−B)]
= 1 2 ⋅ ( c o s A ⋅ c o s B − s i n A ⋅ s i n B + c o s A ⋅ c o s B + s i n A ⋅ s i n B ) =\dfrac{1}{2}\cdot(\mathrm{cos}A\cdot\mathrm{cos}B-\mathrm{sin}A\cdot\mathrm{sin}B+\mathrm{cos}A\cdot\mathrm{cos}B+\mathrm{sin}A\cdot\mathrm{sin}B) =21⋅(cosA⋅cosB−sinA⋅sinB+cosA⋅cosB+sinA⋅sinB)
= c o s A ⋅ c o s B =\mathrm{cos}A\cdot\mathrm{cosB} =cosA⋅cosB
公式3
s i n A ⋅ c o s B = 1 2 ⋅ [ s i n ( A + B ) + s i n ( A − B ) ] \mathrm{sinA}\cdot\mathrm{cos}B=\dfrac{1}{2}\cdot[\mathrm{sin}(A+B)+\mathrm{sin}(A-B)] sinA⋅cosB=21⋅[sin(A+B)+sin(A−B)].
1 2 ⋅ [ s i n ( A + B ) + s i n ( A − B ) ] \dfrac{1}{2}\cdot[\mathrm{sin}(A+B)+\mathrm{sin}(A-B)] 21⋅[sin(A+B)+sin(A−B)]
= 1 2 ⋅ ( s i n A ⋅ c o s B + c o s A ⋅ s i n B + s i n A ⋅ c o s B − c o s A ⋅ s i n B ) =\dfrac{1}{2}\cdot(\mathrm{sin}A\cdot\mathrm{cos}B+\mathrm{cos}A\cdot\mathrm{sin}B+\mathrm{sin}A\cdot\mathrm{cos}B-\mathrm{cos}A\cdot\mathrm{sin}B) =21⋅(sinA⋅cosB+cosA⋅sinB+sinA⋅cosB−cosA⋅sinB)
= s i n A ⋅ c o s B =\mathrm{sinA}\cdot\mathrm{cos}B =sinA⋅cosB
公式4
c o s A ⋅ s i n B = 1 2 ⋅ [ s i n ( A + B ) − s i n ( A − B ) ] \mathrm{cos}A\cdot\mathrm{sin}B=\dfrac{1}{2}\cdot[\mathrm{sin}(A+B)-\mathrm{sin}(A-B)] cosA⋅sinB=21⋅[sin(A+B)−sin(A−B)].
1 2 ⋅ [ s i n ( A + B ) − s i n ( A − B ) ] \dfrac{1}{2}\cdot[\mathrm{sin}(A+B)-\mathrm{sin}(A-B)] 21⋅[sin(A+B)−sin(A−B)]
= 1 2 ⋅ ( s i n A ⋅ c o s B + c o s A ⋅ s i n B − s i n A ⋅ c o s B + c o s A ⋅ s i n B ) =\dfrac{1}{2}\cdot(\mathrm{sin}A\cdot\mathrm{cos}B+\mathrm{cos}A\cdot\mathrm{sin}B-\mathrm{sin}A\cdot\mathrm{cos}B+\mathrm{cos}A\cdot\mathrm{sin}B) =21⋅(sinA⋅cosB+cosA⋅sinB−sinA⋅cosB+cosA⋅sinB)
= c o s A ⋅ s i n B =\mathrm{cos}A\cdot\mathrm{sin}B =cosA⋅sinB
和差化积公式
公式1
s i n A + s i n B = 2 ⋅ s i n A + B 2 ⋅ c o s A − B 2 . \mathrm{sin}A+\mathrm{sin}B=2\cdot\mathrm{sin}\dfrac{A+B}{2}\cdot\mathrm{cos}\dfrac{A-B}{2}. sinA+sinB=2⋅sin2A+B⋅cos2A−B.
2 ⋅ s i n A + B 2 ⋅ c o s A − B 2 2\cdot\mathrm{sin}\dfrac{A+B}{2}\cdot\mathrm{cos}\dfrac{A-B}{2} 2⋅sin2A+B⋅cos2A−B
= 2 ⋅ ( s i n A 2 ⋅ c o s B 2 + c o s A 2 ⋅ s i n B 2 ) ⋅ ( c o s A 2 ⋅ c o s B 2 + s i n A 2 ⋅ s i n B 2 ) =2\cdot(\mathrm{sin}\dfrac{A}{2}\cdot\mathrm{cos}\dfrac{B}{2}+\mathrm{cos}\dfrac{A}{2}\cdot\mathrm{sin}\dfrac{B}{2})\cdot(\mathrm{cos}\dfrac{A}{2}\cdot\mathrm{cos}\dfrac{B}{2}+\mathrm{sin}\dfrac{A}{2}\cdot\mathrm{sin}\dfrac{B}{2}) =2⋅(sin2A⋅cos2B+cos2A⋅sin2B)⋅(cos2A⋅cos2B+sin2A⋅sin2B)
= 2 ⋅ ( s i n A 2 c o s A 2 c o s 2 B 2 + c o s 2 A 2 s i n B 2 c o s B 2 + s i n 2 A 2 s i n B 2 c o s B 2 + s i n 2 B 2 s i n A 2 c o s A 2 ) =2\cdot(\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{A}{2}\mathrm{cos}^2\dfrac{B}{2}+\mathrm{cos}^2\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}\mathrm{cos}\dfrac{B}{2}+\mathrm{sin}^2\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}\mathrm{cos}\dfrac{B}{2}+\mathrm{sin}^2\dfrac{B}{2}\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{A}{2}) =2⋅(sin2Acos2Acos22B+cos22Asin2Bcos2B+sin22Asin2Bcos2B+sin22Bsin2Acos2A)
= 2 s i n A 2 c o s A 2 ⋅ ( s i n 2 B 2 + c o s 2 B 2 ) + 2 s i n B 2 c o s B 2 ⋅ ( s i n 2 A 2 + c o s 2 A 2 ) =2\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{A}{2}\cdot(\mathrm{sin}^2\dfrac{B}{2}+\mathrm{cos}^2\dfrac{B}{2})+2\mathrm{sin}\dfrac{B}{2}\mathrm{cos}\dfrac{B}{2}\cdot(\mathrm{sin}^2\dfrac{A}{2}+\mathrm{cos}^2\dfrac{A}{2}) =2sin2Acos2A⋅(sin22B+cos22B)+2sin2Bcos2B⋅(sin22A+cos22A)
= s i n A + s i n B =\mathrm{sin}A+\mathrm{sin}B =sinA+sinB
公式2
s i n A − s i n B = 2 ⋅ c o s A + B 2 ⋅ s i n A − B 2 . \mathrm{sin}A-\mathrm{sin}B=2\cdot\mathrm{cos}\dfrac{A+B}{2}\cdot\mathrm{sin}\dfrac{A-B}{2}. sinA−sinB=2⋅cos2A+B⋅sin2A−B.
2 ⋅ c o s A + B 2 ⋅ s i n A − B 2 2\cdot\mathrm{cos}\dfrac{A+B}{2}\cdot\mathrm{sin}\dfrac{A-B}{2} 2⋅cos2A+B⋅sin2A−B
= 2 ⋅ ( c o s A 2 c o s B 2 − s i n A 2 s i n B 2 ) ⋅ ( s i n A 2 c o s B 2 − c o s A 2 s i n B 2 ) =2\cdot(\mathrm{cos}\dfrac{A}{2}\mathrm{cos}\dfrac{B}{2}-\mathrm{sin}\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2})\cdot(\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{B}{2}-\mathrm{cos}\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}) =2⋅(cos2Acos2B−sin2Asin2B)⋅(sin2Acos2B−cos2Asin2B)
= 2 ⋅ ( s i n A 2 c o s A 2 c o s 2 B 2 − s i n 2 A 2 s i n B 2 c o s B 2 − c o s 2 A 2 s i n B 2 c o s B 2 + s i n A 2 c o s A 2 s i n 2 B 2 ) =2\cdot(\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{A}{2}\mathrm{cos}^2\dfrac{B}{2}-\mathrm{sin}^2\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}\mathrm{cos}\dfrac{B}{2}-\mathrm{cos}^2\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}\mathrm{cos}\dfrac{B}{2}+\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{A}{2}\mathrm{sin}^2\dfrac{B}{2}) =2⋅(sin2Acos2Acos22B−sin22Asin2Bcos2B−cos22Asin2Bcos2B+sin2Acos2Asin22B)
= 2 s i n A 2 c o s A 2 ⋅ ( s i n 2 B 2 + c o s 2 B 2 ) − 2 s i n B 2 c o s B 2 ⋅ ( s i n 2 A 2 + c o s 2 A 2 ) =2\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{A}{2}\cdot(\mathrm{sin}^2\dfrac{B}{2}+\mathrm{cos}^2\dfrac{B}{2})-2\mathrm{sin}\dfrac{B}{2}\mathrm{cos}\dfrac{B}{2}\cdot(\mathrm{sin}^2\dfrac{A}{2}+\mathrm{cos}^2\dfrac{A}{2}) =2sin2Acos2A⋅(sin22B+cos22B)−2sin2Bcos2B⋅(sin22A+cos22A)
= s i n A − s i n B =\mathrm{sin}A-\mathrm{sin}B =sinA−sinB
公式3
c o s A + c o s B = 2 ⋅ c o s A + B 2 c o s A − B 2 \mathrm{cos}A+\mathrm{cos}B=2\cdot\mathrm{cos}\dfrac{A+B}{2}\mathrm{cos}\dfrac{A-B}{2} cosA+cosB=2⋅cos2A+Bcos2A−B.
2 ⋅ c o s A + B 2 c o s A − B 2 2\cdot\mathrm{cos}\dfrac{A+B}{2}\mathrm{cos}\dfrac{A-B}{2} 2⋅cos2A+Bcos2A−B
= 2 ⋅ ( c o s A 2 c o s B 2 − s i n A 2 s i n B 2 ) ⋅ ( c o s A 2 c o s B 2 + s i n A 2 s i n B 2 ) =2\cdot(\mathrm{cos}\dfrac{A}{2}\mathrm{cos}\dfrac{B}{2}-\mathrm{sin}\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2})\cdot(\mathrm{cos}\dfrac{A}{2}\mathrm{cos}\dfrac{B}{2}+\mathrm{sin}\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}) =2⋅(cos2Acos2B−sin2Asin2B)⋅(cos2Acos2B+sin2Asin2B)
= 2 c o s 2 A 2 c o s 2 B 2 − 2 s i n 2 A 2 s i n 2 B 2 =2\mathrm{cos}^2\dfrac{A}{2}\mathrm{cos}^2\dfrac{B}{2}-2\mathrm{sin}^2\dfrac{A}{2}\mathrm{sin}^2\dfrac{B}{2} =2cos22Acos22B−2sin22Asin22B
= 2 c o s 2 A 2 ( 1 − s i n 2 B 2 ) − 2 s i n 2 B 2 ( 1 − c o s 2 A 2 ) =2\mathrm{cos}^2\dfrac{A}{2}(1-\mathrm{sin}^2\dfrac{B}{2})-2\mathrm{sin}^2\dfrac{B}{2}(1-\mathrm{cos}^2\dfrac{A}{2}) =2cos22A(1−sin22B)−2sin22B(1−cos22A)
= 2 c o s 2 A 2 − 2 c o s 2 A 2 s i n 2 B 2 − 2 s i n 2 B 2 + 2 c o s 2 A 2 s i n 2 B 2 =2\mathrm{cos}^2\dfrac{A}{2}-2\mathrm{cos}^2\dfrac{A}{2}\mathrm{sin}^2\dfrac{B}{2}-2\mathrm{sin}^2\dfrac{B}{2}+2\mathrm{cos}^2\dfrac{A}{2}\mathrm{sin}^2\dfrac{B}{2} =2cos22A−2cos22Asin22B−2sin22B+2cos22Asin22B
= 2 c o s 2 A 2 − 1 + 1 − 2 s i n 2 B 2 =2\mathrm{cos}^2\dfrac{A}{2}-1+1-2\mathrm{sin}^2\dfrac{B}{2} =2cos22A−1+1−2sin22B
= c o s A + c o s B =\mathrm{cos}A+\mathrm{cos}B =cosA+cosB
公式4
c o s A − c o s B = − 2 s i n A + B 2 s i n A − B 2 \mathrm{cos}A-\mathrm{cos}B=-2\mathrm{sin}\dfrac{A+B}{2}\mathrm{sin}\dfrac{A-B}{2} cosA−cosB=−2sin2A+Bsin2A−B.
− 2 s i n A + B 2 s i n A − B 2 -2\mathrm{sin}\dfrac{A+B}{2}\mathrm{sin}\dfrac{A-B}{2} −2sin2A+Bsin2A−B
− 2 ⋅ ( s i n A 2 c o s B 2 + c o s A 2 s i n B 2 ) ⋅ ( s i n A 2 c o s B 2 − c o s A 2 s i n B 2 ) -2\cdot(\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{B}{2}+\mathrm{cos}\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2})\cdot(\mathrm{sin}\dfrac{A}{2}\mathrm{cos}\dfrac{B}{2}-\mathrm{cos}\dfrac{A}{2}\mathrm{sin}\dfrac{B}{2}) −2⋅(sin2Acos2B+cos2Asin2B)⋅(sin2Acos2B−cos2Asin2B)
= 2 c o s 2 A 2 s i n 2 B 2 − 2 s i n 2 A 2 c o s 2 B 2 =2\mathrm{cos}^2\dfrac{A}{2}\mathrm{sin}^2\dfrac{B}{2}-2\mathrm{sin}^2\dfrac{A}{2}\mathrm{cos}^2\dfrac{B}{2} =2cos22Asin22B−2sin22Acos22B
= 2 c o s 2 A 2 ( 1 − c o s 2 B 2 ) − 2 c o s 2 B 2 ( 1 − c o s 2 A 2 ) =2\mathrm{cos}^2\dfrac{A}{2}(1-\mathrm{cos}^2\dfrac{B}{2})-2\mathrm{cos}^2\dfrac{B}{2}(1-\mathrm{cos}^2\dfrac{A}{2}) =2cos22A(1−cos22B)−2cos22B(1−cos22A)
= 2 c o s 2 A 2 − 2 c o s 2 A 2 c o s 2 B 2 − 2 c o s 2 B 2 + 2 c o s 2 A 2 c o s 2 B 2 =2\mathrm{cos}^2\dfrac{A}{2}-2\mathrm{cos}^2\dfrac{A}{2}\mathrm{cos}^2\dfrac{B}{2}-2\mathrm{cos}^2\dfrac{B}{2}+2\mathrm{cos}^2\dfrac{A}{2}\mathrm{cos}^2\dfrac{B}{2} =2cos22A−2cos22Acos22B−2cos22B+2cos22Acos22B
= 2 c o s 2 A 2 − 1 + 1 − 2 c o s 2 B 2 =2\mathrm{cos}^2\dfrac{A}{2}-1+1-2\mathrm{cos}^2\dfrac{B}{2} =2cos22A−1+1−2cos22B
= c o s A − c o s B =\mathrm{cos}A-\mathrm{cos}B =cosA−cosB