Tangents of polar curves

Tangents of polar curves

In this lesson, we will learn how to find the tangent line of polar curves. Just like how we can find the tangent of Cartesian and parametric equations, we can do the same for polar equations. First, we will examine a generalized formula to taking the derivative, and apply it to finding tangents. Then we will look at a few examples to finding the first derivative. Lastly, we will do some applications which involve finding tangent lines of polar curves at a specified point.
Basic Concepts: Power rule, Derivative of trigonometric functions , Derivative of exponential functions, Polar coordinates
Related Concepts: Slope and equation of tangent line

Lessons

In order to find the tangent line to polar curves, we have to take the derivative in polar coordinates.
Here is the formula to take the derivative in polar coordinates:
dydx=drdθsin⁡θ+rcos⁡θdrdθcos⁡θ−rsin⁡θ\frac{dy}{dx}=\frac{\frac{dr}{d \theta}\sin \theta+r\;\cos \theta}{\frac{dr}{d \theta}\cos \theta-r\;\sin \theta}
  • Introduction
    Tangents of Polar Curves Overview
  • 1.
    Finding the Derivative
    Find dydx\frac{dy}{dx} for each of the following polar equations:
    a)
    r=sin⁡θ+θr=\sin \theta + \theta

    b)
    r=sin⁡θcos⁡θ r= \frac{\sin \theta}{\cos \theta}

  • 2.
    Finding the Tangent Line
    Find the tangent line with the following polar curves at the specified point:
    r=sin⁡(3θ)r=\sin (3\theta) at θ=π4\theta = \frac{\pi}{4}
  • 3.
    Finding the Tangent Line
    Find the tangent line with the following polar curves at the specified point:
    r=θcos⁡θ r=\theta \cos \theta at θ=0\theta =0
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6.
Parametric Equations and Polar Coordinates
6.1
Defining curves with parametric equations
6.2
Tangent and concavity of parametric equations
6.3
Area of parametric equations
6.4
Arc length and surface area of parametric equations
6.5
Polar coordinates
6.6
Tangents of polar curves
6.7
Area of polar curves
6.8
Arc length of polar curves
Integral Calculus