What is the equation of the line that is normal to the polar curve #f(theta)=theta- sin((5theta)/2-pi/2)+tan((2theta)/3-pi/2) # at #theta = pi#?

Answer 1

#y = 11/(18pi+6sqrt3)x +1/18#

From the reference Tangents with Polar Coordinates we obtain the equation:

#dy/dx = ((dr)/(d theta)sin(theta)+rcos(theta))/((dr)/(d theta)cos(theta)-rsin(theta))" [1]"#
The slope of the tangent line, #m_t,# is equation [1] evaluated at #theta = pi#:
#m_t = r/((dr)/(d theta))" [2]"#
The slope of the normal line, #m_n# is the negative reciprocal of equation [2]:
#m_n = -((dr)/(d theta))/r" [3]"#

We are given:

#r= f(theta)=theta- sin((5theta)/2-pi/2)+tan((2theta)/3-pi/2) #
Evaluated at #theta = pi#:
#r= pi + sqrt(3)/3 = (3pi+sqrt3)/3#
Compute #(dr)/(d theta)#:
#(dr)/(d theta) = 1 - 5/2cos((5theta)/2-pi/2)+2/3sec^2((2theta)/3-pi/2)#
Evaluated at #theta = pi#:
#-11/18#
#m_n = 11/(18pi+6sqrt3)#
Compute the Cartesian point from the polar point #(r,theta)=((3pi+sqrt3)/3,pi)#:
#x = rcos(theta)#
#x = (3pi+sqrt3)/3cos(pi)#
#x = -(3pi+sqrt3)/3#
#y = rsin(theta)#
#y = (3pi+sqrt3)/3sin(pi)#
#y = 0#
Use the point slope form, #y = m(x - x_1)+y_1# for the equation of a line:
#y = 11/(18pi+6sqrt3)(x - (-(3pi+sqrt3)/3))+ 0#

Simplify to slope-intercept form:

#y = 11/(18pi+6sqrt3)x +1/18#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7