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Home Trigonometric Functions Slopes And Curve Sketching in Polar Coordinates Examples Example 2: Sketching r = sin 2θ
 
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Example 2: Sketching r = sin 2θ

Sketch the curve r = sin 2θ.

Step 1

dr/dθ = 2 cos 2θ.

Step 2

r = 0 at θ = 0, 07_trigonometric_functions-472.gif, π, 07_trigonometric_functions-473.gif. dr/dθ = 0 at θ = 07_trigonometric_functions-474.gif

Step 3

See Figure 7.8.8.07_trigonometric_functions-475.gif

Figure 7.8.8

Step 4

We take values at intervals of 07_trigonometric_functions-476.gif beginning at θ = 0. We can save some time by observing that the values from π to 2π are the same as those from 0 to π.

θ

r = sin 2θ

dr/dθ

tan ψ

Comments

0 and π

0

2

0

crosses origin

π/8 and 9π/8

√2/2

√2

1/2

|r| increasing

2π/8 and 10π/8

1

0

-

max

3π/8 and 11π/8

√2/2

-√2

-1/2

|r| decreasing

4π/8 and 12π/8

0

-2

0

crosses origin

5π/8 and 13π/8

-√2/2

-√2

1/2

|r| increasing

6π/8 and 14π/8

-1

0

-

min

7π/8 and 15π/8

-√2/2

√2

-1/2

|r| decreasing

Step 5

We plot the points and trace out the curve as θ increases from 0 to 2π. Figure 7.8.9 shows the curve at various stages of development. The graph looks like a four-leaf clover.07_trigonometric_functions-477.gif

Figure 7.8.9

 

If r approaches ∞ as θ approaches 0 or π, the curve may have a horizontal asymptote which can be found by computing the limit of y. At θ = π/2 or 3π/2 there may be vertical asymptotes. The method is illustrated in the following example.
Home Trigonometric Functions Slopes And Curve Sketching in Polar Coordinates Examples Example 2: Sketching r = sin 2θ

Last Update: 2006-11-15