Question

Ajutați-mă, vă rog!!! Vreau cu explicație dacă se poate!!
Exprimați în funcție de sin x sau cos x:
a) sin(-25•π/2+x)
b) cos(15•π/2+x)

Answer 1

$\sin\Big(\dfrac{\pi}{2} - \alpha\Big) = \cos \alpha\\ \\ \cos\Big(\dfrac{\pi}{2} - \alpha\Big) = \sin \alpha \\ \\ \sin(2k\pi + \alpha) = \sin\alpha ,\quad k \in \mathbb{Z}\\ \\\cos(2k\pi + \alpha) = \cos\alpha ,\quad k \in \mathbb{Z}\\ \\ \sin(-x) = -\sin x\\ \\ \cos(-x) = \cos x\\ \\ \\\sin\Big(-\dfrac{25pi}{2}+x\Big)= \sin\Big(-\dfrac{24\pi}{2} - \dfrac{\pi}{2} + x\Big) = \\ \\ =\sin\Big(-2\cdot 12 \pi - \dfrac{\pi}{2} + x) = -\sin \Big(2\cdot 12\pi + \dfrac{\pi}{2} - x \Big) =$
$\\ \\ = -\sin \Big(\dfrac{\pi}{2} -x\Big) = -\cos x$


$\cos\Big(\dfrac{15\pi}{2}+x\Big) = \cos\Big(\dfrac{16\pi}{2} - \dfrac{\pi}{2}+x\Big) = \cos\Big(2\cdot 4\pi - \dfrac{\pi}{2} +x\Big) = \\ \\ = \cos\Big(- \dfrac{\pi}{2} + x\Big) = \cos\Big(\dfrac{\pi}{2} - x\Big) = \sin x$