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Trigonometry Confused about a couple of trig. identities.

skatenerd

Active member
Oct 3, 2012
114
I've got this problem right now, which asks me to prove that
$$Ccos(\omega_{o}t-\phi)=Asin(\omega_{o}t)+Bcos(\omega_{o}t)$$
This proved to be a bit more difficult than I expected, so I looked up a complete list of trig identities.
$$cos(a\pm{b})=cos(a)cos(b)\mp{sin(a)sin(b)}$$
seems like the only one that could be helpful in my situation, however when I try to think of a way where the original equation makes sense, I am really not able to convince myself.
Wouldn't the only way for the original equation to make sense be if
\(sin(\phi)=cos(\phi)=1\)? As far as I know this is only possible for \(\frac{-3\pi}{4}\).
Kind of stuck here, any help would be very appreciated
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Re: confused about a couple trig identities.

I've got this problem right now, which asks me to prove that
$$Ccos(\omega_{o}t-\phi)=Asin(\omega_{o}t)+Bcos(\omega_{o}t)$$
This proved to be a bit more difficult than I expected, so I looked up a complete list of trig identities.
$$cos(a\pm{b})=cos(a)cos(b)\mp{sin(a)sin(b)}$$
seems like the only one that could be helpful in my situation, however when I try to think of a way where the original equation makes sense, I am really not able to convince myself.
Wouldn't the only way for the original equation to make sense be if
\(sin(\phi)=cos(\phi)=1\)? As far as I know this is only possible for \(\frac{-3\pi}{4}\).
Kind of stuck here, any help would be very appreciated
I am 90% positive I have shown this in my classical mechanics notes in the forum's notes section. They aren't completed so it should be easy to find.
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
Re: confused about a couple trig identities.

I've got this problem right now, which asks me to prove that
$$Ccos(\omega_{o}t-\phi)=Asin(\omega_{o}t)+Bcos(\omega_{o}t)$$
This proved to be a bit more difficult than I expected, so I looked up a complete list of trig identities.
$$cos(a\pm{b})=cos(a)cos(b)\mp{sin(a)sin(b)}$$
seems like the only one that could be helpful in my situation, however when I try to think of a way where the original equation makes sense, I am really not able to convince myself.
Wouldn't the only way for the original equation to make sense be if
\(sin(\phi)=cos(\phi)=1\)? As far as I know this is only possible for \(\frac{-3\pi}{4}\).
Kind of stuck here, any help would be very appreciated
I agree with what you are thinking. Using the identity you have been given

[tex]\displaystyle \begin{align*} C\cos{ \left( \omega_ot - \phi \right)} &= C \left[ \cos{ \left( \omega_ot \right)}\cos{ \left( \phi \right) } + \sin{\left( \omega_ot \right) } \sin{\left( \phi \right) } \right] \\ &= C\sin{\left( \phi \right) } \sin{\left( \omega_ot \right) } + C\cos{\left( \phi \right)} \cos{\left( \omega_ot \right) } \\ &= A\sin{\left( \omega_ot \right) } + B \sin{ \left( \omega_ot \right) } \end{align*}[/tex]

where [tex]\displaystyle \begin{align*} A = C\sin{(\phi)} \end{align*}[/tex] and [tex]\displaystyle \begin{align*} B = C\cos{(\phi)} \end{align*}[/tex]. Here you're not expected to be able to evaluate these constants, you're just expected to show that there ARE some constants that exist which would satisfy your identity.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: confused about a couple trig identities.

$\LaTeX$ tip:

Precede trigonometric/logarithmic functions with a backslash so that their names are not italicized as if they are strings of variables. For example:

sin(\theta) produces $sin(\theta)$

\sin(\theta) produces $\sin(\theta)$
 

DreamWeaver

Well-known member
Sep 16, 2013
337
Re: Confused about a couple trig identities.

Hi Skatenerd! :D

Have you come across the Auxiliary Angle formulae - from trigonometry - before...?


\(\displaystyle A\cos x+ B\sin x=\sqrt{A^2+B^2}\cos(x\pm\varphi)\quad ; \, \varphi=\tan^{-1}\left(\mp \frac{B}{A}\right) \)


\(\displaystyle A\cos x+ B\sin x=\sqrt{A^2+B^2}\sin(x\pm\varphi)\quad ; \, \varphi=\tan^{-1}\left(\pm \frac{A}{B}\right) \)





Incidentally, I just posted a thread in the Puzzles Board that might be of interest...

Auxiliary angle proof