Trigonometric identity forced oscillations

[Gyroscope]

Homework Statement

http://www.jyu.fi/kastdk/olympiads/2004/Theoretical%20Question%203.pdf

http://www.jyu.fi/kastdk/olympiads/2004/Solution%203.pdf

Question A- (b)

They use some trigomentric identity that I don't understand, which one is it?

Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[cristo]

Well, you've got an expression that's in the form sinAsinB; so you look to use the formula cos(A-B)=cosAcosB+sinAsinB.

From this, you know that [tex]\cos\{\omega_it-\phi-\omega t\}=\cos\{\omega_i t-\phi\}\cos(\omega t)+\sin\{\omega_i t-\phi\}\sin(\omega t)[/tex]. You can work out the similar expression that [tex]\cos\{\omega_it-\phi+\omega t\}=\cos\{\omega_i t-\phi\}\cos(\omega t)-\sin\{\omega_i t-\phi\}\sin(\omega t)[/tex] Then subtract the second from the first.

 

[m2003]

Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. These identities are used to simplify and manipulate expressions involving these functions. In the context of forced oscillations, trigonometric identities can be used to represent the motion of an oscillating system in terms of sine and cosine functions.

In the provided link, the trigonometric identity being used is the sum-to-product identity, which states that the sum of two trigonometric functions can be expressed as a product of two other trigonometric functions. This identity is used in the solution to convert the given expression into a simpler form, making it easier to solve for the unknown variables.

In conclusion, trigonometric identities play an important role in solving problems involving oscillatory motion, and the sum-to-product identity is one example of a useful identity in this context. It is important for scientists to have a good understanding of these identities in order to analyze and interpret data related to oscillatory systems.