# sin (180 - c) = sin c how ?

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• Apr 1st 2011, 06:51 AM
saha.subham
sin (180 - c) = sin c how ?
i was going through an example in my book and cam across a step
a + b + c = 180
a+ b+ = 180 - c
sin ( a + b) = sin (180 - c) = sin c

i am not being able to understand how
sin (180 - c) = sin c
can anyone please help me?? is it a printing mistake?? thanks in advance
• Apr 1st 2011, 06:56 AM
Quacky
No, it isn't a printing mistake. The sin curve is periodic and fluctuates between -1 and 1 every 360 degrees, causing oscillations which can be divided into symmetrical sections.

See the graph of y=sin(x) here
• Apr 1st 2011, 07:00 AM
skeeter
Quote:

Originally Posted by saha.subham
i was going through an example in my book and cam across a step
a + b + c = 180
a+ b+ = 180 - c
sin ( a + b) = sin (180 - c) = sin c

i am not being able to understand how
sin (180 - c) = sin c
can anyone please help me?? is it a printing mistake?? thanks in advance

$\sin(180-c) = \sin(180)\cos(c) - \cos(180)\sin(c) = (0) \cdot \cos(c) - (-1) \cdot \sin(c) = \sin(c)$
• Apr 1st 2011, 07:00 AM
topsquark
Quote:

Originally Posted by saha.subham
i was going through an example in my book and cam across a step
a + b + c = 180
a+ b+ = 180 - c
sin ( a + b) = sin (180 - c) = sin c

i am not being able to understand how
sin (180 - c) = sin c
can anyone please help me?? is it a printing mistake?? thanks in advance

Another way:
$sin(a - b) = sin(a)~cos(b) - sin(b)~cos(a)$

In your case
$sin(180 - c) = sin(180)~cos(c) - sin(c)~cos(180)$

$= (0)~cos(c) - sin(c)~(-1) = 0 - (-sin(c)) = sin(c)$

-Dam
• Apr 1st 2011, 09:30 AM
HallsofIvy
Or- using the "circular definition" of sine, sin(t) is the y coordinate of the point (x, y) an angle t, measured counter-clockwise from the positive x-axis. along the circumference of the unit circle. sin(180- x) would be measured back, clockwise from the negative x-axis. By the symmetry of the circle, those two points have the same y value (and their x coordinates have opposite sign- cos(180- c)= -cos(c)).
• Apr 1st 2011, 09:43 AM
skeeter
does a sketch help?