Need help on a cos/sin explanation problem!

**pyrosilver** · January 20th 2009, 07:12 PM

Explain why the value of [cos $theta$ , sin $theta$ ) *dot* [(cos(90+ $theta$ ), sin(90+ $theta$ )] is independent of $theta$ .

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^^ this is one of the problems I have due tomorrow. Problem is I don't even know what it's asking. independent? Are they saying that if you dot those, you can remove the theta and it still works? If this is what its asking I have no idea. i've been thinking about this for like half an hour and i think I'm missing something key? Please help =(

**Prove It** · January 20th 2009, 07:18 PM

Originally Posted by pyrosilver

Explain why the value of [cos $theta$ , sin $theta$ ) *dot* [(cos(90+ $theta$ ), sin(90+ $theta$ )] is independent of $theta$ .

--

^^ this is one of the problems I have due tomorrow. Problem is I don't even know what it's asking. independent? Are they saying that if you dot those, you can remove the theta and it still works? If this is what its asking I have no idea. i've been thinking about this for like half an hour and i think I'm missing something key? Please help =(

Take the dot product and remember that

$\cos{(x - y)} = \cos{x}\cos{y} + \sin{x}\sin{y}$ .

You should find that you get something that doesn't involve a $\theta$ .

**Mush** · January 20th 2009, 07:19 PM

Originally Posted by pyrosilver

Explain why the value of [cos $theta$ , sin $theta$ ) *dot* [(cos(90+ $theta$ ), sin(90+ $theta$ )] is independent of $theta$ .

--

^^ this is one of the problems I have due tomorrow. Problem is I don't even know what it's asking. independent? Are they saying that if you dot those, you can remove the theta and it still works? If this is what its asking I have no idea. i've been thinking about this for like half an hour and i think I'm missing something key? Please help =(

If this is supposed to be the dot product of two vectors $(\cos(\theta), \sin(\theta))$ and $(\cos(\theta + 90), \sin(\theta + 90))$ , then:

$(\cos(\theta), \sin(\theta)) \cdot (\cos(\theta + 90), \sin(\theta + 90))$ $= \cos(\theta)\times\cos(\theta + 90)+sin(\theta)\times \sin(\theta + 90)$

$= \cos(\theta)(\cos(\theta)\cos(90)-\sin(\theta)\sin(90))+sin(\theta)( \sin(\theta)\cos(90) + \sin(90)\cos(\theta))$

Since $\sin(90) = 1$ and $\cos(90) = 0$ :

$= \cos(\theta)(-\sin(\theta))+sin(\theta)\cos(\theta)$

$= -\cos(\theta)\sin(\theta)+sin(\theta)\cos(\theta) = 0$

You should expect this result, since the 2nd vector is the same as the first vector, except it has been rotated through and angle of 90 degrees, and is hence perpendicular. And the dot product of two perpendicular vectors is always 0.

**Prove It** · January 20th 2009, 07:25 PM

Originally Posted by Mush

If this is supposed to be the dot product of two vectors $(\cos(\theta), \sin(\theta))$ and $(\cos(\theta + 90), \sin(\theta + 90))$ , then:

$(\cos(\theta), \sin(\theta)) \cdot (\cos(\theta + 90), \sin(\theta + 90))$ $= \cos(\theta)\times\cos(\theta + 90)+sin(\theta)\times \sin(\theta + 90)$

$= \cos(\theta)(\cos(\theta)\cos(90)-\sin(\theta)\sin(90))+sin(\theta)( \sin(\theta)\cos(90) + \sin(90)\cos(\theta))$

Since $\sin(90) = 1$ and $\cos(90) = 0$ :

$= \cos(\theta)(-\sin(\theta))+sin(\theta)\cos(\theta)$

$= -\cos(\theta)\sin(\theta)+sin(\theta)\cos(\theta) = 0$

You should expect this result, since the 2nd vector is the same as the first vector, except it has been rotated through and angle of 90 degrees, and is hence perpendicular. And the dot product of two perpendicular vectors is always 0.

My way is easier :P

**pyrosilver** · January 20th 2009, 07:29 PM

Thanks both of you =) I understand it now!

**Mush** · January 20th 2009, 08:18 PM

Originally Posted by Prove It

My way is easier :P

Yeah. Way easier!

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