3sin(x+(1/3)Pi) = sin(x)+2sin(x+(1/3)Pi)+cos(x+(1/6)Pi)
I am sure it is true, they both have the same graph and my Maple 14 software tell me it is true when I issue validity check command. I just want algebraic proof.
I am sure it is true, they both have the same graph and my Maple 14 software tell me it is true when I issue validity check command. I just want algebraic proof.
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3sin(x+1/3pi) = 2sin(x+1/3pi) + sin(x +1/3pi)
= 2sin(x+1/3pi) + ((sinx cos1/3pi + cosx sin pi/3)
= 2sin(x + 1/3pi) + 1/2 sinx + cosx sin pi/3
(But, sin pi/3 can be written as 2 cos pi/6 sin pi/6):
= 2sin(x+1/3pi) + 1/2 sinx + 2 cosx cos pi/6 sin pi/6
= 2 sin (x+1/3pi) + 1/2 sin x + cos x cos pi/6
= 2 sin(x+1/3pi) + cosx cos pi/6 - 1/2sinx + sin x
= 2 sin (x+1/3pi) + cosx cospi/6 - sinx sinpi/6 + sinx
= 2 sin(x + pi/3) + cos(x+pi/6) + sinx
= 2sin(x+1/3pi) + ((sinx cos1/3pi + cosx sin pi/3)
= 2sin(x + 1/3pi) + 1/2 sinx + cosx sin pi/3
(But, sin pi/3 can be written as 2 cos pi/6 sin pi/6):
= 2sin(x+1/3pi) + 1/2 sinx + 2 cosx cos pi/6 sin pi/6
= 2 sin (x+1/3pi) + 1/2 sin x + cos x cos pi/6
= 2 sin(x+1/3pi) + cosx cos pi/6 - 1/2sinx + sin x
= 2 sin (x+1/3pi) + cosx cospi/6 - sinx sinpi/6 + sinx
= 2 sin(x + pi/3) + cos(x+pi/6) + sinx