   Chapter 10.4, Problem 106E

Chapter
Section
Textbook Problem

# Finding an Angle In Exercises 107-112, use the result of Exercise 106 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ . Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of θ . Identify the angle ψ .Polar Equation Value of θ r = 3 ( 1 − cos θ ) θ = 3 π 4

To determine

To calculate: The angle ψ between the radial and tangent line for the polar equation r=3(1cosθ) at θ=3π4 and then sketch its graph with the help of graphing utility along with its radial and tangent line.

Explanation

Given:

The polar equation is given as r=3(1cosθ) at θ=3π4.

Formula used:

The angle ψ between the radial and tangent line to the graph for the polar equation r=f(θ) is given by,

tanψ=rdrdθ

Calculation:

Consider the polar equation, r=3(1cosθ).

Differentiate r with respect to θ,

drdθ=ddθ(3(1cosθ))=ddθ(33cosθ)=3sinθ

To calculate tanψ, use the formula,

tanψ=rdrdθ=3(1cosθ)3sinθ=(1cosθ)sinθ

Substitute θ=3π4, the value of ψ is,

tanψ=1cos3π4sin3π4=1

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