With reference to the previous exercise, prove that if then there exist with such that

* Proof. * Let

Then, and . Further, we have,

since

Hence, we know there exists such that

But, if , then

Thus,

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Stumbling Robot

A Fraction of a Dot
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Prove any linear combination of sine and cosine can be expressed solely in terms of sine

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With reference to the previous exercise, prove that if then there exist with such that

* Proof. * Let

Then, and . Further, we have,

since

Hence, we know there exists such that

But, if , then

Thus,

How do we know that there exists alpha? Shouldn’t we prove the continuity of cosine or something like that?

I agree.

I think your solution assumed for granted that the codomain of sin(x) is [0,1]. Which has no analytic proof in prior. So I think this is not a valid proof. Do you have any way around this?

The codomain of sin(x) is [-1,1] cause the Pythagorean identity sin^2(theta)+cos^2(theta)=1 and so his solution is legit.