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Consider data consisting of pairs $(y_i, x_i)$ for $i = 1,..,n$ arising from a non-linear regression model:

$y_i = \alpha + \beta\sin(\omega x_i + \delta) + \epsilon_i$ where $\epsilon_i \sim N(0,\sigma^2)$

and $\alpha, \beta, \omega, \delta,$ and $\sigma^2$ are all unknown parameters.

This can be rewritten as:

$y_i = \alpha + \gamma_1\sin(\omega x_i) + \gamma_2\cos(\omega x_i) + \epsilon_i$ where $\epsilon_i \sim N(0,\sigma^2)$

How do I get from one to the other, and what is the relationship between $(\gamma_1,\gamma_2)$ and $(\beta,\delta)$?

whuber
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Malachias Invictus
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    1. In the first line where you have $i_i$ do you mean $x_i$? 2. If this is a Bayesian question you should tag accordingly. 3. note that we can't tell you what *your* priors are -- there's no "should" for priors. 4. "*How do I get from one to the other*" is already answered [here](http://stats.stackexchange.com/questions/60500/how-to-find-a-good-fit-for-semi-sinusoidal-model-in-r/60504#60504) (this is the problem with scattergun questions; you should try to stick to one issue at a time).... please edit your question accordingly. – Glen_b Mar 20 '17 at 08:22
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    This is answered at https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities. – whuber Mar 20 '17 at 16:31
  • if it is a homework please add self-study tag – Haitao Du Mar 20 '17 at 16:37
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    I restored the [tag:trigonometry] tag because that is precisely what your question is about--it's not really about nonlinear modeling or regression. Perhaps that extra hint will help you find the answer yourself. – whuber May 07 '18 at 21:43
  • Exactly. Your answer is on the trigonometry. – dankernler May 08 '18 at 01:29

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