Graph of r=sin 2θ | Lecture Note - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2019
Views:

375

Pages:

2
Author:

Tania Mcknight

Graph of r = sin 2θ This curve is a favorite; a similar curve appears in the homework. We’ll plot a few points (sin 2θ, θ) to get an idea of what the graph of this curve looks like. θ r = sin 2θ 0 0 pi 1 4 pi 0 2 Note that sin 2θ > 0 for 0 < θ < π2 . So the curve starts at the origin, goes √ √ outward to the point (1, π4 ) (which is ( 22 , 22 ) in rectangular coordinates), then returns to the origin as θ moves to point “up”. Figure 1: Graph of r = sin 2θ for 0 < θ < π 2. Because of the symmetries of the sine function, the curve will do something similar in each quadrant. However, it’s useful to watch the curve being drawn in order to understand how its parts are connected. The “loops” in the graph are caused by r = sin 2θ changing sign each time the graph intersects the origin. When π2 < θ < π, our angle is in the second quadrant; the portion of the graph corresponding to those values of θ appears opposite the angle, in the fourth quadrant. If we want to compute the area of one petal of this rose, we have to be careful to use the right bounds for θ. 7 2 1 (0 < θ < π4 ) 8 5 4 6 3 Figure 2: Graph of r = sin 2θ; a four leaf rose.

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