Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.r=8 sin 0 and r= 8 cos 0Select the correct answer below, and if necessary, fill in the answer box to complete your choice.(Type an ordered pair. Type exact answers for each coordinate, using t as needed. Type the coordinate for 0 in radians between 0 and T. Use a comma to separate answers as needed.)O A. The intersection points do not consist of the pole, but the points.O B. The intersection points consist of the pole andO C. There are no intersection points.

Answered: Use algebraic methods to find as many…

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter8: Complex Numbers And Polarcoordinates

Section8.6: Equations In Polar Coordinates And Their Graphs

Problem 4PS

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Question

Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.
r=8 sin 0 and r= 8 cos 0
Select the correct answer below, and if necessary, fill in the answer box to complete your choice.
(Type an ordered pair. Type exact answers for each coordinate, using t as needed. Type the coordinate for 0 in radians between 0 and T. Use a comma to separate answers as needed.)
O A. The intersection points do not consist of the pole, but the points.
O B. The intersection points consist of the pole and
O C. There are no intersection points.

Expert Solution

Step 1

To Determine: use algebraic methods to find as many intersection points of the following curves as possible.Use graphical methods to identify the remaining intersection points.

Given: we have two curves

$r = 8 \sin θ a n d r = 8 \cos θ$

Explanation: we have two curves

$r = 8 \sin θ - - - - - - - - - - (1) a n d r = 8 \cos θ - - - - - - - - - - (2)$

these are polar curves and now we will find the intersection points by equation both the equations 1 and 2 as follows.

$8 \sin θ = 8 \cos θ \Rightarrow \frac{8 \sin θ}{8 \cos θ} = 1 \Rightarrow \tan θ = 1 \Rightarrow θ = \frac{π}{4} a n d \frac{5 π}{4}$

so on the interval $[0, 2 π]$ ,the function will intersect at

$θ = \frac{π}{4} a n d θ = \frac{5 π}{4}$

and at these values we have

$r = 8 \sin \frac{π}{4} = \frac{8}{\sqrt{2}} = \frac{8 \sqrt{2}}{2} = 4 \sqrt{2}$ and $r = 8 \cos \frac{π}{4} = \frac{8}{\sqrt{2}} = \frac{8 \sqrt{2}}{2} = 4 \sqrt{2}$

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