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Question Number 211075 by ajfour last updated on 27/Aug/24

$$\sin (\theta +\varphi )=\theta$$$$\sin \theta =\varphi$$$$find\theta and\varphi .$$

Answered by Ghisom last updated on 27/Aug/24

$$\mathrm{we}\mathrm{can}\mathrm{only}\mathrm{approximate}$$$$\varphi =\sin \theta$$$$\sin (\theta +\sin \theta )=\theta$$$$1.\theta =\varphi =0$$$$2.\theta \approx \pm .973696169714\wedge \varphi \approx \pm .826969516935$$

Answered by ajfour last updated on 27/Aug/24

$$\cos (\theta +\varphi )=\sqrt{1-{\theta }^2}$$$$\tan (\theta +\varphi )=\frac{\theta }{\sqrt{1-{\theta }^2}}$$$$\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }=\frac{\theta }{\sqrt{1-{\theta }^2}}$$$$\tan \varphi =\frac{\theta -\sqrt{1-{\theta }^2}\tan \theta }{\sqrt{1-{\theta }^2}+\theta \tan \theta }$$$$...yessobit$$

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