   Chapter 10.4, Problem 108E

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 = 4 sin 2 θ θ = π 6

To determine

To calculate: The angle ψ between the radial and tangent line for the polar equation r=4sin2θ at θ=π6 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=4sin2θ at θ=π4.

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=4sin2θ.

Differentiate r with respect to θ,

drdθ=ddθ(4sin2θ)=4cos2θ2=8cos2θ

To calculate tanψ, use the formula

tanψ=rdrdθ=4sin2θ8cos2θ=12tan2θ

Substitute θ=π6, the value of ψ is,

tanψ=12tan2(π6

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