Please help
How do you graph the polar equation
r=8-sec#theta# ?

Question

Josiah Anderson · Accepted Answer

See the explanation and graphs.

As sec values #notin ( - 1, 1 )#, #r = 8 - sec theta notin ( -1+ 8, 1 + 8 ) = ( 7, 9 )# r is periodic,with period #2pi#. Short Table, for $theta in #theta in [ 0, pi ]#, sans asymptotic #pi/2 and 3/2pi#: #( r, theta )#: ( 7, 0 ) ( 6.845, pi/6 ) ( 6, pi/3 ) ( oo, pi/2 )# #( 9.155, 5/6pi ) ( 10, 2pi/3 ) ( 9, pi )#. #r = 0, at theta =1.4455# rad. The graph is symmetrical about #theta = 0#. graph is outside the annular ( circular ) region # 7 < r < 9 ). Converting to the Cartesian form, using #( x, y ) = r ( cos theta, sin theta ) and 0 <= sqrt ( x^2 + y^2 ) = r#, # (x^2 + y^2)^0.5(1+1/x) = 8 #, The Socratic graph is immediate, with asymptote x = 0. graph{((x^2 + y^2)^0.5(1+1/x) - 8)(x+0.01y)=0[-40 40 -20 20]} See the bounding circles #r = 7 and r= 9#. graph{((x^2 + y^2)^0.5(1+1/x) - 8)(x^2+y^2-49)(x^2+y^2-81)=0[-20 20 -10 10]}

Delilah Aguilar · Answer

undefined To graph the polar equation $ r = 8 - \sec(\theta) $, follow these steps:

1. Plot the asymptotes: The equation $ r = 8 - \sec(\theta) $ has vertical asymptotes wherever $ \sec(\theta) $ is undefined. Since $ \sec(\theta) $ is undefined when $ \cos(\theta) = 0 $, there will be vertical asymptotes where $ \cos(\theta) = 0 $, which are $ \theta = \frac{\pi}{2} $ and $ \theta = \frac{3\pi}{2} $.

2. Determine the behavior near the asymptotes: As $ \theta $ approaches $ \frac{\pi}{2} $ or $ \frac{3\pi}{2} $, $ \sec(\theta) $ approaches $ +\infty $, so $ r $ approaches $ -\infty $. Therefore, the graph will approach the vertical asymptotes as $ r $ decreases without bound.

3. Plot key points: Choose some values of $ \theta $ and compute the corresponding values of $ r $ to plot key points on the graph.

4. Connect the points: Connect the plotted points smoothly to sketch the graph.

Remember, the graph of polar equations may not always be perfect due to the limitations of plotting tools. However, plotting several points and understanding the behavior near asymptotes will help you visualize the graph accurately.

Please help How do you graph the polar equation r=8-sec#theta# ?