sin(a+h)-sin a/h

Question

how can i solve this??

Answer 1

sin(a+h)=sin(a)cos(h)+cos(a)sin(h).

(sin(a+h)-sin(a))/h=(sin(a)cos(h)+cos(a)sin(h)-sin(a))/h=

(sin(a)(cos(h)-1)+cos(a)sin(h))/h.

That's the difference quotient.

Watch what happens when h is very small:

sin(h)≈h when h is small, and cos(h)≈1.

(sin(a)(cos(h)-1)+cos(a)sin(h))/h≈(sin(a)(1-1)+hcos(a))/h=(0+hcos(a))/h=cos(a).

cos(a) is the derivative of sin(a).