how-to-justify-that-sin-x-7pi-2-cos-x-

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)