Chapter 10.4, Problem 106E

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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(Î¸).

and tanÏˆ=|rdrdÎ¸| and the graph is shown as below:

Formula used:

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

dydx=dydÎ¸dxdÎ¸=f(Î¸)cosÎ¸+fâ€²(Î¸)sinÎ¸âˆ’f(Î¸)sinÎ¸+fâ€²(Î¸)cosÎ¸.

Use the trigonometric formula,

tan(x+y)=tanx+tany1âˆ’tanxtany

Proof:

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

dydx=f(Î¸)cosÎ¸+fâ€²(Î¸)sinÎ¸âˆ’f(Î¸)sinÎ¸+fâ€²(Î¸)cosÎ¸ â€¦â€¦ (1)

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

tan(Î¸+Ïˆ)=tanÎ¸+tanÏˆ1âˆ’tanÎ¸tanÏˆ

Since, tanÎ¸=sinÎ¸cosÎ¸,

tan(Î¸+Ïˆ)=sinÎ¸+cosÎ¸tanÏˆcosÎ¸âˆ’sinÎ¸tanÏˆ â€¦â€¦ (2)

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

f(Î¸)cosÎ¸+fâ€²(Î¸)sinÎ¸âˆ’f(Î¸

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