Can someone explain how to do this?

Question

Prove:
sin (pi-x)= sin (x) using the addition identity

And
how to find (without a calculator):
sin 40 degrees and cos 40 degrees if sin 20 degrees is approximately equal to 3 and cos 20 degrees is approximately equal to .9


I don't need the answer just a little bit about how to do the problems. Thank You

Answer 1

sin (pi - x) = sin x.
sin (pi - x) = (sin pi)(cos x) - (cos pi)(sin x).
sin pi = 0 and cos pi = -1.
therefore, sin (pi - x) = sin x.

===================================

sin 40 = sin (2*20) = 2sin 20cos 20.
cos 40 = cos (2*20) = (2cos^2 20) - 1.

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Answer 2

sin(40) degrees is sin( 20 + 20) degrees
and sin (20 + 20) = sin(20) x cos(20) + cos(20) x sin(20)
or 2 x sin(20) x cos(20)
so , approx, sin (40) degrees = 2(0.3)(0.9) = 2(0.27)= 0.54
The actual value is 0.6423 ---- there is a large error here because the approximations for sin(20) and cos(20) only go to one decimal place.

Answer 3

sin(a-b) = sinacosb -cosasinb
sin(pi-x) = sin pi cosx -cos pi sinx
sin(pi-x) = 0 * cosx - (-1)*sinx = sin x

Sin (20+20) = sin 2*20 = 2sin20cos20 = 2(.3)(.9) = .54
Cos(20+20) = cos^2 20 -sin^2 20 = .81- .09 = .72