+26367 Find the equation of the asymptote of the graph of \(r = \cos ( 2\theta) \sec( \theta)\).
I assume the function is a polar function:
https://www.wolframalpha.com/input/?i=polar+plot+r%3Dcos%282x%29sec%28x%29
The asymptote of the graph is \(\mathbf{x = -1}\).
source: https://www.youtube.com/watch?v=7Ae3VHPyMMQ
\(\begin{array}{|rcll|} \hline \mathbf{r} &=& \mathbf{\cos (2\theta) \sec( \theta)} \\\\ r &=& \dfrac{\cos (2\theta)} {\cos( \theta)} \quad | \quad \theta=\dfrac{\pi}{2} \Rightarrow r\to \infty \\\\ && \text{Set $\alpha = \dfrac{\pi}{2}$ } \\\\ \dfrac{dr}{d\theta } &=& \dfrac{-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) } {\cos^2(\theta)} \\\\ \dfrac{d\theta }{dr} &=& \dfrac{\cos^2(\theta)} {-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) } \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} r^2\dfrac{d\theta}{dr}\\ \\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(2\theta)} {\cos^2( \theta)} \left(\dfrac{\cos^2(\theta)} {-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) }\right) \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(2\theta)} {-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) } \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(2\dfrac{\pi}{2})} {-2\sin(2\dfrac{\pi}{2})\cos(\dfrac{\pi}{2})+\cos(2\dfrac{\pi}{2})\sin(\dfrac{\pi}{2}) } \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(\pi)} {-2\sin(\pi)\cos(\dfrac{\pi}{2})+\cos(\pi)\sin(\dfrac{\pi}{2}) } \\\\ p &=& \dfrac{(-1)^2} {-2\cdot 0\cdot 0+(-1)(1)} \\\\ p &=& \dfrac{1} {-1} \\\\ \mathbf{p} &=& \mathbf{-1} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{p} &=& \mathbf{r \sin(\alpha-\theta)} \quad | \quad r=-1,\ \alpha = \dfrac{\pi}{2} \\\\ -1 &=& r \sin(\dfrac{\pi}{2} -\theta) \\\\ -1 &=& r \cos( \theta) \\\\ \mathbf{r} &=&\mathbf{ -\dfrac{1}{\cos( \theta)} \quad | \quad \text{the asymptote of the graph }} \\ \hline \end{array}\)
polar plot asymptote see: https://www.wolframalpha.com/input/?i=polar+plot+r%3D+-1%2Fcos%28x%29
Asymptote in Cartesian Coordinates:
To convert from Polar Coordinates (r,\(\theta\)) to Cartesian Coordinates (x,y) :
\(x = r \times \cos(\theta) \\ y = r \times \sin(\theta)\)
\(\begin{array}{|rcll|} \hline \mathbf{r} &=&\mathbf{ -\dfrac{1}{\cos( \theta)} } \quad | \quad \cos{\theta}=\dfrac{x}{r} \\\\ r &=& -\dfrac{1}{ \dfrac{x}{r} } \\ r &=& -\dfrac{r}{ x } \\ xr &=& -r \quad | \quad :r \\\\ \mathbf{x} &=& \mathbf{-1 } \\ && \mathbf{\text{the asymptote of the graph in Cartesian Coordinates}} \\ \hline \end{array}\)
hint: The function \(r = \cos ( 2\theta) \sec( \theta)\) in cartesian coordinates: \((x^2+y^2) (1+x) = 2x^2 \quad | \quad r^2 = x^2+y^2,\ x = r \times \cos(\theta) ,\ y = r \times \sin(\theta) \)
+26367 Find the equation of the asymptote of the graph of \(r = \cos 2 \theta \sec \theta\).
I assume the function is a polar function:
polar plot see: https://www.wolframalpha.com/input/?i=polar+plot+r%3Dcos%282x%29sec%28x%29
The asymptote of the graph is \(\mathbf{x = -1}\).
Source: https://www.youtube.com/watch?v=7Ae3VHPyMMQ
\(\begin{array}{|rcll|} \hline \mathbf{r} &=& \mathbf{\cos (2\theta) \sec( \theta)} \\\\ r &=& \dfrac{\cos (2\theta)} {\cos( \theta)} \quad | \quad \theta=\dfrac{\pi}{2} \Rightarrow r\to \infty \\\\ && \text{Set $\alpha = \dfrac{\pi}{2}$ } \\\\ \dfrac{dr}{d\theta } &=& \dfrac{-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) } {\cos^2(\theta)} \\\\ \dfrac{d\theta }{dr} &=& \dfrac{\cos^2(\theta)} {-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) } \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} r^2\dfrac{d\theta}{dr}\\ \\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(2\theta)} {\cos^2( \theta)} \left(\dfrac{\cos^2(\theta)} {-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) }\right) \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(2\theta)} {-2\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta) } \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(2\dfrac{\pi}{2})} {-2\sin(2\dfrac{\pi}{2})\cos(\dfrac{\pi}{2})+\cos(2\dfrac{\pi}{2})\sin(\dfrac{\pi}{2}) } \\\\ p &=& \lim \limits_{\theta\to \dfrac{\pi}{2}} \dfrac{\cos^2(\pi)} {-2\sin(\pi)\cos(\dfrac{\pi}{2})+\cos(\pi)\sin(\dfrac{\pi}{2}) } \\\\ p &=& \dfrac{(-1)^2} {-2\cdot 0\cdot 0+(-1)(1)} \\\\ p &=& \dfrac{1} {-1} \\\\ \mathbf{p} &=& \mathbf{-1} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{p} &=& \mathbf{r \sin(\alpha-\theta)} \quad | \quad r=-1,\ \alpha = \dfrac{\pi}{2} \\\\ -1 &=& r \sin(\dfrac{\pi}{2} -\theta) \\\\ -1 &=& r \cos( \theta) \\\\ \mathbf{r} &=&\mathbf{ -\dfrac{1}{\cos( \theta)} \quad | \quad \text{the asymptote of the graph }} \\ \hline \end{array}\)
polar plot asymtote see: https://www.wolframalpha.com/input/?i=polar+plot+r%3D+-1%2Fcos%28x%29
Asymptote in Cartesian Coordinates:
To convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :
\(x = r \times \cos( \theta ) \\ y = r \times \sin( \theta )\)
\(\begin{array}{|rcll|} \hline \mathbf{r} &=&\mathbf{ -\dfrac{1}{\cos( \theta)} } \quad | \quad \cos{\theta}=\dfrac{x}{r} \\\\ r &=& -\dfrac{1}{ \dfrac{x}{r} } \\ r &=& -\dfrac{r}{ x } \\ xr &=& -r \quad | \quad :r \\\\ \mathbf{x} &=& \mathbf{-1 } \\ && \mathbf{\text{the asymptote of the graph in Cartesian Coordinates}} \\ \hline \end{array}\)
