trigonometry (symmetry arguments and trig identities)

[ripple]

Hey guys,
Got this question on an assignment and have no idea what to do.

1. Show that the following relationships are true
(i) cos(θ − 180) = −cos(θ) using symmetry arguments
(ii) sin(θ − 180) = −sin(θ) using symmetry arguments
(iii) Confirm the relationships in parts (i) and (ii) using trigonometric identities

I have no idea how to use symmetry arguments or how to present them.
Also (iii) asks me to confirm these relationships using trig identities, I thought they were already using trig identities.
Thanks for any help in advance

 

[ksdhart2]

I'm not familiar with the term "symmetry arguments," but my best guess is that it's talking about how sine is an odd function so -sin(t) = sin(-t) for any t, and cosine is an even function so cos(t) = cos(-t) for any t. If this is not correct, please reply with the definition as given by your class/textbook/instructor. For now, I'll assume my definition is correct...

To work the proofs, I'd begin by noting that an extension of the symmetry arguments tells you that sin(t) = -sin(-t). If we let $\displaystyle t=\theta+180^{\circ}$, then we have $\displaystyle cos(\theta+180^{\circ})=cos(-(\theta+180^{\circ})) \implies cos(\theta+180^{\circ})=cos(-\theta-180^{\circ})$ and $\displaystyle sin(\theta + 180^{\circ})=-sin(-(\theta+180^{\circ})) \implies sin(\theta + 180^{\circ})=-sin(-\theta - 180^{\circ})$. Where do those lead you?

Then for the third part, I agree with your assessment that they're already identities in their own right, but I suspect you're being asked to proof them by starting from more "basic" identities. In this case, you might use $\displaystyle sin(\theta+90^{\circ})=cos(\theta)$ and $\displaystyle cos(\theta+90^{\circ})=-sin(\theta))$, along with $\displaystyle sin(a+b)=sin(a) cos(b) + cos(a) sin(b)$ and $\displaystyle cos(a+b)=cos(a) cos(b) - sin(a) sin(b)$. When using these identities, keep in mind that $\displaystyle \theta+180^{\circ}=(\theta+90^{\circ})+90^{\circ}$. Where does all of this lead you?

 

[Deleted member 4993]

Hey guys,
Got this question on an assignment and have no idea what to do.

1. Show that the following relationships are true
(i) cos(θ − 180) = −cos(θ) using symmetry arguments
(ii) sin(θ − 180) = −sin(θ) using symmetry arguments
(iii) Confirm the relationships in parts (i) and (ii) using trigonometric identities

I have no idea how to use symmetry arguments or how to present them.
Also (iii) asks me to confirm these relationships using trig identities, I thought they were already using trig identities.
Thanks for any help in advance

(iii) Confirm the relationships in parts (i) and (ii) using trigonometric identities

I think you would need to use:

sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B) &

sin(180°) = 0 & cos(180°) = -1