Use the Shifting Theorem to find the Laplace transform

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The discussion centers on confusion regarding a step in the application of the shifting theorem to find the Laplace transform, specifically related to the cosine identity. Participants express frustration over numerous typos in the provided solutions, which have contributed to misunderstandings. The criticism highlights that these errors are consistent across multiple threads related to the same course on ordinary differential equations (ODEs). The quality of the educational materials is deemed inadequate, leading to embarrassment for the institution involved. Overall, the conversation emphasizes the need for clearer and error-free instructional content.
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Homework Statement
Please see below
Relevant Equations
Shifting theorem
For (b),
1713665142390.png

I'm confused on the highlighted step. Does someone please explain to me how they got from the left to the right?

Thanks!
 
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cos(a-b)=cos(a)cos(b)+sin(a)sin(b) and cos(π/4)=sin(π/4)=1/√2
 
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Note again the typos in the solution. For example the ##1/\sqrt 2## multiplying only one of the terms. The amount of typos in your problems is really beyond critisism. It is outright embarrassing to whatever institution is giving these out.
 
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Orodruin said:
Note again the typos in the solution. For example the ##1/\sqrt 2## multiplying only one of the terms. The amount of typos in your problems is really beyond critisism. It is outright embarrassing to whatever institution is giving these out.
Criticism.
 
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WWGD said:
Criticism.
Valid critisism. I have no idea where OP is studying, but I know that in every single problem they have posted the provided solution has been full of typos. Many times those typos have been the source of OP’s confusion.
 
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Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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