As illustrated in the accompanying figure, let P (r, ) be apoint on the polar curve r =

Chapter 10, Problem 61

(choose chapter or problem)

As illustrated in the accompanying figure, let \(P(r, \theta)\) be a point on the polar curve \(r=f(\theta)\), let \(\Psi\) be the smallest counterclockwise angle from the extended radius OP to the tangent line at P, and let \(\varphi\) be the angle of inclination of the tangent line. Derive the formula

\(\tan \Psi=\frac{r}{d r / d \theta}\)

by substituting \(\tan \varphi\) for \(d y / d x\) in Formula (2) and applying the trigonometric identity

\(\tan (\varphi-\theta)=\frac{\tan \varphi-\tan \theta}{1+\tan \varphi \tan \theta}\)

Equation Transcription:

Text Transcription:

P(r, theta)

r=f(theta)

psi

phi

tan psi = r/dr/d theta

tan phi

dy/dx

tan(phi - theta)=tan phi - tan theta/1 + tan phi tan theta

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