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I encountered this form of equation in a functions review exercise that uses the function $P(t) = 82.5 - 67.5 \cos (π t/6) + t$. The question was as follows: At what value $t$ will $P$ be equal to $200$ for the very first time?

So, I tried to solve it and ended with the following equation $67.5 \cos (π t/6)= t-117.5$.

However, I don't know what to do next and ended up looking to Desmos for answers. But is there an actual general solution to this equation, or better yet a solution to the trig-linear hybrid equation I posted at the top as an example? Thank you.

Gonçalo
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Gabriel Sebastian
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    There is no analytical solution, only numerical. You can check when is the first maximum that is over 200. A maximum occurs slightly after the cosine term is $-1$. So you want $82.5+67.5+t>200$ and $\frac \pi6 t=(2n+1)\pi$. This will give you $n=4$ and $t=54$ – Andrei Dec 20 '24 at 14:05
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    This is a Kepler equation – Тyma Gaidash Dec 20 '24 at 14:41

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Your equation can be easily changed into : $$\cos(nx) = \frac{p}{k}-\frac{l}{nk}(nx)$$ Substituting y as nx, we get: $$\cos(y) = \frac{p}{k}-\frac{l}{nk}(y)$$

I believe that this equation has no "closed form" solution, as it is very similar to the equation $\cos(x) = x$, which has been discussed in this post:What is the solution of $\cos(x)=x$?

Aside from that, it is easy to achieve a approximation by using Newton's Method or perhaps expanding cos(y) into its Taylor series (without the aid of graphing calculators).

Will Chang
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