Another way to find trig identities

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The discussion explores an elegant method for finding trigonometric identities using specific mathematical identities. Participants inquire about the connection between complex exponentials and sinusoidal functions, suggesting that further reading on the topic may be beneficial. The thread is ultimately locked, with a note indicating that a related question has been moved to the Precalculus Homework section. The focus remains on the mathematical principles behind the identities rather than personal opinions. This highlights the importance of understanding the underlying connections in trigonometry.
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Homework Statement
Please see below
Relevant Equations
Please see below
Using the identity's
1678912539056.png


(1)
1678912560583.png

(2)
1678912570481.png

Gives,
1678912607008.png

Why dose this elegant method work?

Many thanks!
 
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Sorry, I posted in the wrong forum. This was meant to be for the precalculus forum.

Many thanks!
 
Are you asking about the connection between complex exponentials and sinusoidals? If so, you may find that this article addresses your question.
 
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Callumnc1 said:
Sorry, I posted in the wrong forum. This was meant to be for the precalculus forum.

Many thanks!
Thread locked. A related question is posted in the Precalc Homework section.
 
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Thread 'Magnitude of buoyant force in fluids of different densities'

The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?
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