What is the equation of the tangent line of #r=-5cos(-theta-(pi)/4) + 2sin(theta-(pi)/12)# at #theta=(-5pi)/12#?

Answer 1

The polar equation is invalid when #theta = -(5pi)/12#

We have a polar equation:

# r = -5cos(-theta-pi/4) + 2sin(theta-pi/12) #
When #theta = -(5pi)/12# we have:
# r = -5cos(pi/6) + 2sin(-pi/2) # # \ \ = -(5sqrt(3))/2-2 # # \ \ ~~-6 #
Hence, The polar equation is invalid when #theta = -(5pi)/12#
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 2

See explanation

Recall that in polar coordinates, you have #x = r*cosθ, y=r*sinθ#. The equation for the slope of tangent line to the curve at any given point is still #dy/dx#, despite being in polar coordinates. Since y and x are functions of r and θ, this means...
#dy/dx = (dy/(dθ))/(dx/(dθ)) = ((dr)/(dθ)sinθ + rcosθ)/((dr)/(dθ)cosθ - rsinθ)#.
Since #r(θ) = -5cos(-θ-pi/4) + 2sin(θ-pi/12), (dr)/(dθ) = -5sin(-θ-pi/4) + 2 cos(θ-pi/12)#...

Then...

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 3

To find the equation of the tangent line at a given point on a polar curve, we first need to express the polar curve in Cartesian coordinates, then find its derivative with respect to theta, and finally evaluate the derivative at the given theta value to determine the slope of the tangent line.

The equation of the polar curve r = -5cos(-θ - π/4) + 2sin(θ - π/12) can be expressed in Cartesian coordinates using the relationships x = rcos(θ) and y = rsin(θ). Substituting these expressions into the given equation, we obtain the Cartesian equation of the curve.

After finding the Cartesian equation, we differentiate it with respect to θ to find the derivative, dy/dx. Then, we evaluate dy/dx at θ = -5π/12 to find the slope of the tangent line.

Once we have the slope of the tangent line, we use the point-slope form of the equation of a line, y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope of the tangent line. Substituting the given point (x1, y1) and the slope m into this equation gives us the equation of the tangent line.

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