4. 3.5/6.5 points All Submissions Notes Question: SerEssen11.AE.06. .Question part Points Submissions 1 2 3 4 5 6 7 8
.5 .5 .5 1 1 0/1 0/1 0/1
1/15 1/15 1/15 1/15 1/15 5/15 0/15 0/15
Total 3.5/6.5 ... Example 1.6 Cartesian and Polar Coordinates
Goal Understand how to convert from plane rectangular coordinatesto plan polar coordinates and vice versa.
Problem (a) The Cartesian coordinates of a point in the xy-planeare (x, y) = (-3.37, -2.73) m, as shown in Active Figure 1.5 Findthe polar coordinates of this point. (b) Convert (r, ?) = (4.61 m,36.5°) to rectangular coordinates.
Strategy Apply the trigonometric functions and their inverses,together with the Pythagorean theorem.
Figure 1.5 Converting from Cartesian coordinates to polarcoordinates.
Solution
(a) Cartesian to Polar
Take the square root of both sides of Equation 1.2 to find theradial coordinate. r =
= 1 m
Use Equation 1.1 for the tangent function to find the angle withthe inverse tangent, adding 180° because the angle is actually inthe third quadrant: tan ? = = = 0.810
? = tan-1 (0.810) = 39.0° + 180°
? = 2°
(b) Polar to Cartesian
Use the trigonometric definitions, Equation 1.1. x = r cos? = (4.61m) cos 36.5°
x = 3 m
y = r sin? = (4.61 m) sin 36.5°
y = 4 m
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Remarks When we take up vectors in two dimensions in Chapter 3, wewill routinely use a similar process to find the direction andmagnitude of a given vector from its components, or, conversely, tofind the components from the vector's magnitude anddirection.
Exercise 1.6 Hints: Getting Started | I'm Stuck
(a) Find the polar coordinates corresponding to (x, y) = (3.11,1.28) m.
r = 5 m
? = 6°
Your response differs significantly from the correct answer. Reworkyour solution from the beginning and check each stepcarefully.
(b) Find the Cartesian coordinates corresponding to (r, ?) = (4.07m, 52.5°)
x = 7 m
y = 8 m
Anything you could do would help, but please explain slowly. Thanksagain. It is for my technical physics 115 class.
