Question

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?

Answer 1

follow the steps

Explanation:

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 :)

Answer 2

Use a diagram and some reasoning...

Explanation:

...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.
Therefore,
`(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`

GOOD LUCK