Question

Using taylors theorem solve
Expand sinxy in power of (x-1) (y- π/2)

Answer 1

$\huge\red{\underline\orange{\underline\purple{Answer:}}}$How do I expand e^xcosy in powers of x and (y-pi/2) using Taylor’s theorem?

If you know the expansion of exex, then it's quite easy to expand excosyexcosy

ex=1+x+x22!+x33!+...ex=1+x+x22!+x33!+...

Now, just put xcosyxcosy in place of xx in the above expression to get:

excosy=1+xcosy+(xcosy)22!+(xcosy)33!+...excosy=1+xcosy+(xcosy)22!+(xcosy)33!+...

Now use the fact that cosy=sin(π/2−y)cosy=sin(π/2−y) so that

excosy=1+xsin(π/2−y)+(xsin(π/2−y))22!+(xsin(π/2−y))33!+...=f(x,π/2−y)excosy=1+xsin(π/2−y)+(xsin(π/2−y))22!+(xsin(π/2−y))33!+...=f(x,π/2−y)

Answer 2

Answer:

If you know the expansion of ex , then it's quite easy to expand excosy

ex=1+x+x22!+x33!+...

Now, just put xcosy in place of x in the above expression to get:

excosy=1+xcosy+(xcosy)22!+(xcosy)33!+...

Now use the fact that cosy=sin(π/2−y) so that

excosy=1+xsin(π/2−y)+(xsin(π/2−y))22!+(xsin(π/2−y))33!+...=f(x,π/2−y)

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