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019.09 points | Previous Answers SerCP9 1.AE.009 EXAMPLE 1.9Cartesian and Polar Coordinates GOAL Understand how to convert from plane rectangular coordinates to plane polar coordinates and vice versa. y (m) PROBLEM (a) The Cartesian coordinates of a point in the xy-plane are (x,y) (-3.50 m, -2.50 m), as shown in the NY figure. Find the polar coordinates of this point. (b) Convert (r, θ) = (5.00 m, 37.0°) to rectangular coordinates x (m) (-3.50,-2.50) STRATEGY Apply the trigonometric functions and their inverses, together with the Pythagorean theorem. Converting from Cartesian coordinates to polar coordinates. SOLUTION (A) Cartesian to Polar conversion Take the square root of both sides of the equation to find the radial coordinate r-vK2*),2 (-3.50 m)2 + (-2.50 m)"- 4.30 tan θ =上一ー2.50 m tan-1(0.714) 0,714 Use the tangent function to find the angle with the inverse tangent, adding 180° because the angle is actually in third quadrant. x -3.50 m 35.50 + 180°-2160 (B) Polar to Cartesian conversion Use the trigonometric definition equations x rcos 6 y rsin (5.00 m) cos 37.03.99 m (5.00 m) sin 37.0° 3.01 m LEARN MORE REMARKS When we take up vectors in two dimensions later in the book, we will routinely use a similar process to find the direction and magnitude of a given vector from its components, or, conversely, to find the components from the vector's magnitude and direction 0

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a) r squareroot (-3.7)^2 + (-2.84)^2 m = 4.66 m tan theta = -2.84 m/-3.7 m = 0.767 theta = tan^-1 (0.767) = 37.5 + 180 degree = 218 degree b) x = r cos theta = (4.99 m) cos(36.8 degree) = 3.99 m y = r sin theta = (4.99 m) sin 36.8 degree = 2.99 m a) r = squareroot 3.38^2 + 1.6^2 m = 3.74 m tan theta = 1.6/3.38 = 0.473 or, theta = tan^-1 (0.473) = 25.3 degree b) x = r cos theta = (4.24 m) cos(52.4 degree) = 2.59 m y = r sin theta = (4.24 m) sin(52.4 degree) = 3.36 m