how to justify that sin (x−((7π)/2) )= cos x


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Question Number 120244 by zahaku last updated on 30/Oct/20

$h o w t o j u s t i f y t h a t \sin (x - \frac{7 π}{2}) = \cos x$
Commented by zahaku last updated on 30/Oct/20

$\sin (x - \frac{7 π}{2}) = - \sin (2 π + (x + \frac{3 π}{2}) ?$
Commented by Aziztisffola last updated on 30/Oct/20

$\frac{7 π}{2} \equiv - \frac{π}{2} (\mod 2 π)$ $x - \frac{7 π}{2} \equiv (x + \frac{π}{2}) (\mod 2 π)$ $\sin (x + \frac{π}{2}) = \cos x$
	Answered by benjo_mathlover last updated on 30/Oct/20

	$\Rightarrow \sin (x - \frac{7 π}{2}) = - \sin (2 π + (x + \frac{3 π}{2}))$ $= - \sin (x + \frac{3 π}{2})$ $= - [\sin (\frac{3 π}{2}) \cos x + \cos (\frac{3 π}{2}) \sin x]$ $= - (- \cos x + 0) = \cos x$
	Answered by TANMAY PANACEA last updated on 30/Oct/20

	$s i n (x - \frac{7 π}{2})$ $= - s i n (\frac{7 π}{2} - x)$ $= - (- c o s x) [h e r e 7 (o d d n u m b e r) \times \frac{π}{2} - x] l i e s i n 4 t h q u a d r a n t s o (- v e) s i g n$ $= c o s x$
	Answered by mathmax by abdo last updated on 30/Oct/20

	$\sin (x - \frac{7 π}{2}) = \sin (x - \frac{8 π - π}{2}) = \sin (x - 4 π + \frac{π}{2})$ $= \sin (x + \frac{π}{2}) = cosx (- 4 π is period)$
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