How to calculate θ when you have sin(θ+α)

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The discussion centers on calculating θ from the equation cos(θ) + √2 sin(θ) = r sin(θ + α), highlighting that this is not a standard trigonometric identity. Participants clarify that the equation has infinitely many solutions for α and r, depending on the choice of α that does not make sin(θ + α) zero. A breakdown of the steps reveals that setting r sin(α) = 1 and r cos(α) = √2 leads to two equations that can be solved for r and α. Ultimately, the solution for θ is derived as arcsin(1/3) through various transformations and substitutions, demonstrating the complexity of the problem. The discussion emphasizes the importance of understanding the parametrization and the intermediate value theorem in solving such equations.
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Homework Statement
The solution to a problem involves calculating theta when you know what sin (theta + alpha) is. Somehow, the solution is able to eliminate alpha so that theta = sin^-1(root 2/root 3) - cos^-1(root 2/root 3)
Relevant Equations
sin(theta + alpha) = root 2/root 3
Hi everyone

I'm having trouble understanding 4c. I'm able to follow most of the solution, but don't know understand the step where

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Could someone please explain this step?

The chapter is about trigonometric identities and formulas, but this doesn't fit any of the formulas given. Thanks

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This is not a trigonometric identity. In fact, it's not even an identity. There's a lot more going on under the hood here, that I actually don't fully understand.

##\cos(\theta)+\sqrt{2}\sin(\theta)=r\sin(\theta+\alpha)## has infinitely many solutions for ##\alpha## and ##r##. In fact, for any choice of ##\alpha## that doesn't make ##\sin(\theta+\alpha)## zero, you can just divide and solve for a choice of ##r## that works. This solution is claiming that there exists one choice of ##r## and ##\alpha## that make this equation work for every choice of ##\theta##. I don't really see why this would be true. It's using that magic value of ##\alpha## to compute ##\theta##

Edit to add: I think I get it a bit more, as far as you can say they ##1\cos(\theta)+\sqrt{2}\sin(\theta) = (r\sin(\alpha)) \cos(\theta)+ (r\cos(\alpha)) \sin(\theta)##, you just want ##r\sin(\alpha)=1## and ##r\cos(\alpha)=\sqrt{2}##. That's two equations in two unknowns, there's at least a decent chance you can get a solution.
 
Are you fine starting with
\cos \theta+\sqrt{2} \sin \theta = \sqrt{2}
? Then
\frac{1}{\sqrt{3}}\cos \theta+\frac{\sqrt{2}}{\sqrt{3}} \sin \theta = \frac{\sqrt{2}}{\sqrt{3}}
\sin \alpha \cos \theta + \cos \alpha \sin \theta = \sin (\theta + \alpha) = \frac{\sqrt{2}}{\sqrt{3}}
where
\alpha = \cos^{-1}\frac{\sqrt{2}}{\sqrt{3}}=\sin^{-1}\frac{1}{\sqrt{3}}
which are equivalent.
So
\theta = \sin^{-1}\frac{\sqrt{2}}{\sqrt{3}}-\sin^{-1}\frac{1}{\sqrt{3}}=\frac{\pi}{2}-2\sin^{-1}\frac{1}{\sqrt{3}}
or in rectangle triangle ##\triangle## ABC
\theta = \angle BAC - \angle ACB=\frac{\pi}{2}-2 \angle ACB
where AB=1, BC=##\sqrt{2}##, CA=##\sqrt{3}## and ##\angle##ABC=R.
 
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Way to hunt a fly with a cannon! :oops:

Denote ##x:=\theta##. From ## \cos ^2x = 2(1-\sin x)^2## conclude ##3\sin ^2x - 4\sin x +1 = 0##. Put ##z:=\sin x##, then either ##z=1/3## or ##z=1##. Since ##0\leqslant x< \pi/2##, one concludes ##x=\arcsin (1/3)##.The part in the solution in OP, where one let's ##\cos \theta +\sqrt{2}\sin \theta = r\sin (\theta+\alpha)## is effectively using the intermediate value theorem. The RHS is parametrised such that it contains ##\theta##. And then further effort is applied to determine the parametrisation. It is sufficient, but unnecessary. On the other hand, it's good for practicing technique.
 
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Thanks for brilliant answer. Just for checking the coincidence with #3,
\cos \theta = \sin(2 \sin^{-1}\frac{1}{\sqrt{3}})=2\frac{1}{\sqrt{3}}\frac{\sqrt{2}}{\sqrt{3}}=\frac{2\sqrt{2}}{3}
\sin\theta = \frac{1}{3}
 
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Another approach at finding a solution for ##\displaystyle \ \frac{\cos\theta}{1-\sin\theta}=\sqrt{2} \ : ##

Square the equation. Then change ##\ \cos^2 \theta \ ## to ##\ 1-\sin^2 \theta \ ## which is equal to ## \
(1+\sin \theta )(1-\sin \theta) ## .

After cancelling a factor of ##\ (1-\sin \theta) \ ## from numerator & denominator we get :

##\displaystyle \frac{1+\sin \theta }{1-\sin \theta}= 2## .

Solving for ## \sin \theta ## we get ## \ \displaystyle \sin \theta = \frac 1 3 \ ## .
 
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I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
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