Maximillian Patterson

2022-04-28

Problem with inversion of the system $u=\frac{\mathrm{cos}\theta}{1-t\mathrm{cos}\theta},\text{}v=\frac{\mathrm{sin}\theta}{1+t\mathrm{sin}\theta}$

eslasadanv3

Beginner2022-04-29Added 20 answers

One way is to invert each equality and then you can solve them for $\mathrm{sin}\theta$ and $\mathrm{cos}\theta$ . Then, use ${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$ to get rid of $\theta$ and solve for t. Having t as a function of u and v, you can find $\mathrm{sin}\theta +\mathrm{cos}\theta$ easily.

haillarip0c9

Beginner2022-04-30Added 23 answers

You have:

$\mathrm{cos}\theta =\frac{u}{1+ut}\text{}\text{and}\text{}\mathrm{sin}\theta =\frac{v}{1-vt}$

Now use${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$

$\frac{{u}^{2}}{{(1+ut)}^{2}}+\frac{{v}^{2}}{{(1-vt)}^{2}}=1$

olving the above equation you will get equation in t and then you can find t as a function of u and v

Now use

olving the above equation you will get equation in t and then you can find t as a function of u and v

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