How do you derive the trigonometric sum and difference formulas for sin, cos, and tan? I.e: How do I derive something like sin(x+y)=sinxcosy+cosxsiny?

2 Answers
Oct 30, 2017

follow the steps


these are compound angle identities :

  1. sin(A+B) #-=# sinAcosB + cosAsinB

from that:

put B=A
therefore: sin (A+A) = sin (2A)

= sinAcosA+cosAsinA

= 2 sinAcosA(thats the double angle formula) <<<<<

it goes on for all the other compound angles..

:try them and let me know if you had any difficulty, i'll be glad to help :)

Oct 30, 2017

Use a diagram and some reasoning...


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...The best math teacher I ever had taught me: memorize as little as possible in mathematics.

Words to live by, as I can never be 100% sure I remember these trig identities correctly. But refer to the diagram:

Angle AOE is the sum of angles x and y.
Furthermore, segment OA has length 1.
#(AD)/(OA) = AD = siny#

Note now that triangles AFD and OFB are similar.

Angle CAD is therefore x

Line Segments #(AC)/(AD) = cosx#

...but we previously deduced that #AD = siny#. Therefore,

#AC = siny * cosx#

Now, note that segment #(OD)/(OA) = OD = cosy#

And you can see from the diagram that #(DE)/(OD) = sinx#.

Therefore segment #DE = sinxcosy#

Segments DE and CB are equal.

Therefore, Segments #CB + AC = sin(x+y) = cosxsiny + sinxcosy#