Question

Identify the curve by finding a Cartesian equation for the curve.

\({{\rm{r}}^{\rm{2}}}{\rm{cos2\theta = 1}}\)

Short answer

\({{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = 1}}\)this is a hyperbola.

Step-by-step solution

Find a Cartesian equation for the curve and use it to identify it.

Given: \({{\rm{r}}^{\rm{2}}}{\rm{cos(2\theta ) = 1}}\)

\({\rm{cos(2\theta )}}\)trig identity

\({\rm{cos(2\theta ) = co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\theta }} \right){\rm{ - si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\theta }} \right)\)

Using the replacement from the previous line, rewrite the equation.

\({{\rm{r}}^{\rm{2}}}\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\theta }} \right){\rm{ - si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\theta }} \right)} \right){\rm{ = 1}}\)

\({{\rm{r}}^{\rm{2}}}{\rm{co}}{{\rm{s}}^{\rm{2}}}\left( {\rm{\theta }} \right){\rm{ - }}{{\rm{r}}^{\rm{2}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}\left( {\rm{\theta }} \right){\rm{ = 1}}\)

Remember that \({\rm{x = rcos\theta \& y = rsin\theta }}\)

\({{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = 1}}\)

sketch the graph.

Therefore, the graph\({{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = 1}}\). This is a hyperbola.