Question

Write true or false.
If $\sin x = \cos y$ then $x + y = {45^ \circ }$

Answer 1

Hint: Try to remember a principal value of x and see the complimentary value of y then add them and see whether we are getting ${45^ \circ }$ or not.
Complete Step by Step Solution:
Let us do it by taking some examples first we know that $\sin \dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2}$ and the same value of cosine exists when $\cos \dfrac{\pi }{6} = \dfrac{{\sqrt 3 }}{2}$
So in this case $\sin \dfrac{\pi }{3} = \cos \dfrac{\pi }{6} = \dfrac{{\sqrt 3 }}{2}$
So if I add them i will get the required result

\therefore x + y\\\ = \dfrac{\pi }{3} + \dfrac{\pi }{6}\\\ = \dfrac{{2\pi + \pi }}{6}\\\ = \dfrac{{3\pi }}{6}\\\ = \dfrac{\pi }{2} \end{array}$$ And we also know that $$\dfrac{\pi }{2} = {90^ \circ }$$ so from here we can clearly see that the statement written is false Note: we also know that $$\sin x = \cos (90 - x)$$ so this also settles it because we can also write $$\cos y = \sin (90 - y)$$ therefore $$\sin x = \sin (90 - y)$$ and now if we remove sin from both the sides we are left with $$x = {90^ \circ } - y$$ i.e., $$x + y = {90^ \circ }$$ So in both the cases the answer remains to be false.