Confused about a couple of trig. identities.

In summary, the conversation is about proving the equation $$Ccos(\omega_{o}t-\phi)=Asin(\omega_{o}t)+Bcos(\omega_{o}t)$$ using the trigonometric identity $$cos(a\pm{b})=cos(a)cos(b)\mp{sin(a)sin(b)}$$. The speakers discuss the possibility of using the identity and conclude that the only way for the original equation to make sense is if \(sin(\phi)=cos(\phi)=1\). They also mention the use of auxiliary angle formulae and provide a link to a related thread.
  • #1
skate_nerd
176
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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
 
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  • #2
Re: confused about a couple trig identities.

skatenerd said:
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.
 
  • #3
Re: confused about a couple trig identities.

skatenerd said:
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.
 
  • #4
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 $\sin(\theta)$
 
  • #5
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...

http://mathhelpboards.com/challenge-questions-puzzles-28/auxiliary-angle-proof-6759-new.html
 

FAQ: Confused about a couple of trig. identities.

1. What are the main trigonometric identities?

The main trigonometric identities are the Pythagorean identities (sin²𝜃 + cos²𝜃 = 1), the reciprocal identities (csc𝜃 = 1/sin𝜃, sec𝜃 = 1/cos𝜃, cot𝜃 = 1/tan𝜃), and the quotient identities (tan𝜃 = sin𝜃/cos𝜃, cot𝜃 = cos𝜃/sin𝜃).

2. Why are trigonometric identities important?

Trigonometric identities are important because they help simplify and solve complex trigonometric equations and expressions. They also provide a deeper understanding of the relationships between the trigonometric functions.

3. How do I prove a trigonometric identity?

To prove a trigonometric identity, you must manipulate the expressions on both sides of the equation using algebraic and trigonometric properties until they are equal. This involves using the identities themselves, as well as basic algebraic principles.

4. How do I remember all the trigonometric identities?

One way to remember the trigonometric identities is to practice using them frequently. You can also create a cheat sheet or mnemonic devices to help you remember them. Another helpful tip is to understand the relationships between the identities and how they are derived from each other.

5. Can I use trigonometric identities in real-world applications?

Yes, trigonometric identities are used in various fields such as engineering, physics, and astronomy. They are especially useful in solving problems involving angles, distances, and proportions.

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