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Home Trigonometric Functions Slopes And Curve Sketching in Polar Coordinates Proof of Theorem 2
 
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Proof of Theorem 2

PROOF OF THEOREM 2

Assume the curve is not vertical at the point P, that is, dx ≠ 0. Since x = r cos θ, y = r sin θ, we have

07_trigonometric_functions-480.gif

07_trigonometric_functions-481.gif

Figure 7.8.12

By the definition of the tangent line L (see Figure 7.8.12),

07_trigonometric_functions-482.gif

Using the addition formulas,

07_trigonometric_functions-483.gif

Thus

07_trigonometric_functions-484.gif

Multiplying out and canceling, we get

07_trigonometric_functions-485.gif

whence

07_trigonometric_functions-486.gif

If the curve is vertical at P we may use the same proof but with dx/dy instead of dy/dx.
Home Trigonometric Functions Slopes And Curve Sketching in Polar Coordinates Proof of Theorem 2

Last Update: 2006-11-05