anonymous one year ago Does the graph of y = sin x + cos x have even symmetry, odd symmetry, or neither?

1. freckles

$f(x)=\sin(x)+\cos(x) \\ \text{ plug \in } -x \\ f(-x)=? \\ \text{ if you get } f(-x)=f(x) \text{ then } f \text{ is even } \\ \text{ if you get } f(-x)=-f(x) \text{ then } f \text{ is odd }$

2. anonymous

f(-x)=sin(-x)+cos(-x) now what? @freckles

3. freckles

use the fact that sin is an odd function and that cos is an even function

4. freckles

that is you are using that sin(-x)=-sin(x) and that cos(-x)=cos(x)

5. anonymous

okay i got cos(x)=f(-x)+sin(x) @freckles

6. freckles

did you add sin(x) on both sides?

7. 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?

8. freckles

or neither?

9. anonymous

@freckles -sin(x)+cos(x) is the opposite of the original function

10. 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

11. 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

12. 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

13. freckles

if we had: $f(x)=x^2+\cos(x) \\ \text{ and we plug in } -x \\ f(-x)=(-x)^2+\cos(-x) \\ f(-x)=x^2+\cos(x) \\ \text{ this would mean } f \text{ is even } \\ \text{ if we had } f(x)=x^3+\sin(x) \\ \text{ and we plug in } -x \\ \text{ we would have } \\ f(-x)=(-x)^3+\sin(-x) \\ f(-x)=-x^3-\sin(x) \\ f(-x)=-(x^3+\sin(x)) \text{ this is the opposite of } f \\ \text{ so this } f \text{ would be odd }$

14. freckles

anyways was just giving examples of an odd and even function just in case you were confused or whatever

15. freckles

let me know if you still don't understand

16. anonymous

no i think i understand now. thank you so much @freckles

17. freckles

np