4/13/2024 0 Comments Polar coordinates graphHowever, using the properties of symmetry and finding key values of θ θ and r r means fewer calculations will be needed. To graph in the polar coordinate system we construct a table of θ θ and r r values. To graph in the rectangular coordinate system we construct a table of x x and y y values. Graphing Polar Equations by Plotting Points Test the equation for symmetry: r = − 2 cos θ. For example, suppose we are given the equation r = 2 sin θ r = 2 sin θ We replace ( r, θ ) ( r, θ ) with ( − r, − θ ) ( − r, − θ ) to determine if the new equation is equivalent to the original equation. In the first test, we consider symmetry with respect to the line θ = π 2 θ = π 2 ( y-axis). To determine the graph of a polar equation. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of r ) r ) By performing three tests, we will see how to apply the properties of symmetry to polar equations. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. Symmetry is a property that helps us recognize and plot the graph of any equation. All points that satisfy the polar equation are on the graph. Recall that the coordinate pair ( r, θ ) ( r, θ ) indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of θ, θ, and extend a ray from the pole (origin) r r units in the direction of θ. Just as a rectangular equation such as y = x 2 y = x 2 describes the relationship between x x and y y on a Cartesian grid, a polar equation describes a relationship between r r and θ θ on a polar grid. (credit: modification of work by NASA/JPL-Caltech) Testing Polar Equations for Symmetry The angle to the positive x-axis (rotating in the typical counter-clockwise fashion) is 120°.Figure 1 Planets follow elliptical paths as they orbit around the Sun. Remember, this is the reference angle not the angle to the positive x-axis. Since we already know the angle exists in the second quadrant, only positive values are being used. Now, we must calculate the angle using the second conversion equation (if you do not recognize the special right triangle). Assuming you do not recognize the triangle, let us view the calculation using the first conversion equation. It is unnecessary to calculate the length of the hypotenuse if you recognize this special right triangle. Here is a diagram of the point in the second quadrant. So, the final answer, written as (r, θ), is…Įxample 2: Convert (-1, √3) from rectangular form to polar form. Since the angle exists in the fourth quadrant, we have to account for the traditional trigonometric angle relative to the positive x-axis with a counter-clockwise motion. Remember, this angle is the reference angle. To get the distance the point is from the origin, which is the r-value, we will use the first conversion equation, like so. Here is the graph of the rectangular point. It is helpful to get a diagram to see what is going on. Now, let us look at two examples to see how these conversions are done.Įxample 1: Convert (5,-3) to polar form, rounded to the nearest tenth. Using knowledge of trigonometry, we can see the tangent of theta is equal to the opposite (y) over adjacent (x) sides, which is the second conversion equation. Since this is a right triangle, we can employ The Pythagorean Theorem, which is the first of the two conversion equations. The relationship between the x, y, and r-variables should be familiar. To understand the genesis of these equations, examine this diagram. To convert from rectangular to polar coordinates requires different equations.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |