   Chapter 10.4, Problem 106E

Chapter
Section
Textbook Problem

Proof Prove that the tangent of the angle ψ ( 0 ≤ ψ ≤ π / 2 ) between the radial line and the tangent line at the point (r, θ ) on the graph of r = f ( θ ) (see figure) is given by tan ψ = | r d r / d θ | q

To determine

To prove:

The expression tanψ=|rdrdθ| if the tangent of angle ψ(0ψπ2) between the radial line and the tangent line at the point (r,θ) on the graph of r=f(θ) is given..

Explanation

Given:

The given function is r=f(θ).

and tanψ=|rdrdθ| and the graph is shown as below:

Formula used:

The slope of tangent line at point (r,θ) is,

dydx=dydθdxdθ=f(θ)cosθ+f(θ)sinθf(θ)sinθ+f(θ)cosθ.

Use the trigonometric formula,

tan(x+y)=tanx+tany1tanxtany

Proof:

Since the slope of the tangent line at point (r,θ) (r,θ) is

dydx=f(θ)cosθ+f(θ)sinθf(θ)sinθ+f(θ)cosθ …… (1)

From the graph, it can be seen that the slope of the tangent line through A and P is,

tan(θ+ψ)=tanθ+tanψ1tanθtanψ

Since, tanθ=sinθcosθ,

tan(θ+ψ)=sinθ+cosθtanψcosθsinθtanψ …… (2)

Thus equate equation (1) and equation (2) and then cross-multiply them,

f(θ)cosθ+f(θ)sinθf(θ

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