anonymous
  • anonymous
Does the graph of y = sin x + cos x have even symmetry, odd symmetry, or neither?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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freckles
  • freckles
\[f(x)=\sin(x)+\cos(x) \\ \text{ plug \in } -x \\ f(-x)=? \\ \text{ if you get } f(-x)=f(x) \text{ then } f \text{ is even } \\ \text{ if you get } f(-x)=-f(x) \text{ then } f \text{ is odd }\]
anonymous
  • anonymous
f(-x)=sin(-x)+cos(-x) now what? @freckles
freckles
  • freckles
use the fact that sin is an odd function and that cos is an even function

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freckles
  • freckles
that is you are using that sin(-x)=-sin(x) and that cos(-x)=cos(x)
anonymous
  • anonymous
okay i got cos(x)=f(-x)+sin(x) @freckles
freckles
  • freckles
did you add sin(x) on both sides?
freckles
  • freckles
\[f(-x)=\sin(-x)+\cos(-x) \\ f(-x)=-\sin(x)+\cos(x) \] is -sin(x)+cos(x) the original function or the opposite of the original function?
freckles
  • freckles
or neither?
anonymous
  • anonymous
@freckles -sin(x)+cos(x) is the opposite of the original function
freckles
  • freckles
\[-\sin(x)+\cos(x) \neq \sin(x)+\cos(x) \text{ the signs on } \sin \text{ differ } \\ -\sin(x)+\cos(x) \neq -(\sin(x)+\cos(x)) \text{ both the signs would have to be differ}\] what I'm saying is that the opposite of sin(x)+cos(x) is -(sin(x)+cos(x)) but we did not get this so it is not odd and also we did not get sin(x)+cos(x) back so we know this is not even
freckles
  • freckles
another way to say -(sin(x)+cos(x)) is -sin(x)-cos(x) this is what I meant by both of the signs needed to be different
freckles
  • freckles
but anyways in summary -sin(x)+cos(x) is neither the same as or the opposite of sin(x)+cos(x) so sin(x)+cos(x) is neither even or odd
freckles
  • freckles
if we had: \[f(x)=x^2+\cos(x) \\ \text{ and we plug in } -x \\ f(-x)=(-x)^2+\cos(-x) \\ f(-x)=x^2+\cos(x) \\ \text{ this would mean } f \text{ is even } \\ \text{ if we had } f(x)=x^3+\sin(x) \\ \text{ and we plug in } -x \\ \text{ we would have } \\ f(-x)=(-x)^3+\sin(-x) \\ f(-x)=-x^3-\sin(x) \\ f(-x)=-(x^3+\sin(x)) \text{ this is the opposite of } f \\ \text{ so this } f \text{ would be odd }\]
freckles
  • freckles
anyways was just giving examples of an odd and even function just in case you were confused or whatever
freckles
  • freckles
let me know if you still don't understand
anonymous
  • anonymous
no i think i understand now. thank you so much @freckles
freckles
  • freckles
np

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