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Home Trigonometric Functions Slopes And Curve Sketching in Polar Coordinates Theorem 2: Direction of a Curve
 
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Theorem 2: Direction of a Curve

The following theorem gives a simple formula for tan ψ when r ≠ 0.

THEOREM 2

Suppose r = f(θ) is a curve in polar coordinates and dr/dθ exists at a point P where r ≠ 0. Let L be the line tangent to the curve at P and let ψ be the angle between OP and L. Then

07_trigonometric_functions-462.gif

If dr/dθ ≠ 0,

07_trigonometric_functions-463.gif

07_trigonometric_functions-464.gif

Figure 7.8.4

DISCUSSION

When r = Q, P is the origin so the line OP and angle ψ are undefined. The formula can be seen intuitively in Figure 7.8.4. Δθ is infinitesimal. As we move from the point P(r, θ) to the point Q(r + Δr, θ + Δθ) on the curve, the change in the direction perpendicular to OP will be very close to r Δθ, so we have

07_trigonometric_functions-465.gif07_trigonometric_functions-466.gif

We shall postpone the proof to the end of this section. We can use Theorem 2 in curve sketching as follows.

(a) In an interval where tan ψ > 0, the curve is going away from the origin as θ increases because dr/dθ has the same sign as r.

07_trigonometric_functions-467.gif

Figure 7.8.5(a)
(b) Where tan ψ < 0, the curve is going toward the origin as θ increases because dr/dθ has the opposite sign as c.

07_trigonometric_functions-468.gif

Figure 7.8.5(b)
(c)  Where r has either a local maximum or minimum and dr/dθ exists, the curve is going in a direction perpendicular to the radius. This is because dr/dθ = 0 so cot ψ = 0.

07_trigonometric_functions-469.gif

Figure 7.8.5(c)

Each of these cases is shown in Figure 7.8.5.
Home Trigonometric Functions Slopes And Curve Sketching in Polar Coordinates Theorem 2: Direction of a Curve

Last Update: 2006-11-05