klepkowy7c

2022-07-26

Use the inverse trigonometric identities to find the exact value of the expression below.
$\mathrm{sin}\left[\frac{\pi }{4}-{\mathrm{cos}}^{-1}\left(-\frac{1}{\sqrt{2}}\right)\right]$

minotaurafe

Expert

Solution:
$\mathrm{sin}\left[\frac{\pi }{4}-{\mathrm{cos}}^{-1}\left(\frac{-1}{\sqrt{2}}\right)\right]\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left[\frac{\pi }{4}-\left(\pi -{\mathrm{cos}}^{-1}\frac{1}{\sqrt{2}}\right)\right]\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left[\frac{-3\pi }{4}+{\mathrm{cos}}^{-1}\frac{1}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left[\frac{3\pi }{4}+\frac{\pi }{4}\right]\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left(\frac{\pi }{2}\right)\phantom{\rule{0ex}{0ex}}=-\mathrm{sin}\frac{\pi }{2}\phantom{\rule{0ex}{0ex}}=-1\phantom{\rule{0ex}{0ex}}\mathrm{sin}\left[\frac{\pi }{4}-{\mathrm{cos}}^{-1}\frac{-1}{\sqrt{2}}\right]=-1$

Do you have a similar question?