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I remember them by this:
计算正弦时：正余余正号相同
计算余弦时：余余正正号相反

我也不贊成背公式
但是背了公式之後
算起三角函數就比較得心應手了
這是我的感覺

--- 寫到:How to remember the formulas of 和差化積 & 積化和差?

my method:
(1) list the formula of 和角公式
sin(A+B)=sinAcosB + cosAsinB
cos(A+B)=cosAcosB - sinAsinB

I note them as:
s ~ sc +cs
c ~ cc -ss

(2) for 和差化積: let p=(A+B)/2; q=(A-B)/2
then
sinA+sinB=2sinp cosq
sinA-sinB=2cosp sinq
cosA+cosB=2cosp cosq
cosA-cosB=-2sinp sinq

I note them as:
s ~ sc +cs
c ~ cc -ss

(3) for 積化和差: let A=p+q; B=p-q

2sinp cosq=sinA+sinB
2cosp sinq=sinA-sinB
2cosp cosq=cosA+cosB
-2sinp sinq=cosA-cosB

I note them as:
s ~ sc +cs
c ~ cc -ss

Sorry for the typing mistake above.
The only way is to do more. The more you use them, the easier you can remember them.
But, why do you want to remember such terrible formulas?
If I were you, I don't think I'll remember them.
There are a lot more for you to remember in other topics.
Just ignore those formulas printed on the paper.

How to remember the formulas of 和差化積 & 積化和差?

my method:
(1) list the formula of 和角公式
sin(A+B)=sinAcosB + cosAsinB
cos(A+B)=cosAcosB - sinAsinB

I note them as:
s ~ sc +cs
c ~ cc -ss

(2) for 和差化積: let p=(A+B)/2; q=(A-B)/2
then
sinA+sinB=2sinp cosq
sinA-sinB=2cosp sinq
cosA+cosB=2cosp cosq
cosA-cosB=-2sinp sinq

I note them as:
s ~ sc +cs
c ~ cc -ss

(3) for 積化和差: let A=p+q; B=p-q

2sinp cosq=sinA+sinB
2cosp sinq=sinA-sinB
2cosp cosq=cosA+cosB
-2sinp sinq=cosA-cosB

I note them as:
s ~ sc +cs
c ~ cc -ss