How Do Trigonometric Identities Connect to Drawings?

[Dainy]

yes well I got the last problem...but I still want to have an idea were the identity siny+sinx=2sin(x+y)/2cos(x+y)/2 came from please someone sort of explain please! thanx

 

[lurflurf]

yes well I got the last problem...but I still want to have an idea were the identity siny+sinx=2sin(x+y)/2cos(x+y)/2 came from please someone sort of explain please! thanx

what you wrote is not an identity consider x=y=pi/2
sin pi/2+sin pi/2=2
2sin pi/2 cos pi/2=0
do you mean
$$\sin(y)+\sin(x)=2\cos(\frac{y-x}{2})\sin(\frac{y+x}{2})$$
if so
start on the left by writing
y=(y+x)/2+(y-x)/2
x=(y+x)/2-(y-x)/2
then expand using
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
and
sin(a-b)=sin(a)cos(b)-cos(a)sin(b)
add like terms

 

[lalbatros]

back to the root

Dainy:

Make sure that you understand why

cos(a-b) = cos(a) cos(b) + sin(a) sin(b)

and that you can illustrate the meaning of this formula with a drawing.

Many other formulas can be derived form the previous by algebra and by other simple trigonometric rules (likes cos(-b)=cos(b), sin(-b)=-sin(b), cos(pi-b)=-cos(b), ... all rules that can be illustrated by a drawing too.).

So the picture is: a few simple principles and definitions, enough algebra, and you are on your own.