For each pair of coordinates below, convert to polar coordinates. Give the answer to one decimal place. Don't use the polar conversion function on your calculator.
11. [tex]P(6,3)[/tex]
A. [tex]\left(-6.7,26.6^{\circ}\right)[/tex]
B. [tex]\left(-6.7,153.4^{\circ}\right)[/tex]
D. [tex]\left(9,26.6^{\circ}\right)[/tex]
B. [tex]\left(6.7,26.6^{\circ}\right)[/tex]
12. [tex]Q(-12,4)[/tex]
A. [tex]\left(-8,-18.4^{\circ}\right)[/tex]
B [tex]\left(12.6,161.6^{\circ}\right)[/tex]
D. [tex]\left(12.6,-161.6^{\circ}\right)[/tex]
E. [tex]\left(-12.6,161.6^{\circ}\right)[/tex]
Answer (1)
Calculate the radial distance r for point P(6, 3) using r = x 2 + y 2 , resulting in r ≈ 6.7 .
Calculate the angle θ for point P(6, 3) using θ = arctan ( x y ) , resulting in θ ≈ 26. 6 ∘ .
Calculate the radial distance r for point Q(-12, 4) using r = x 2 + y 2 , resulting in r ≈ 12.6 .
Calculate the angle θ for point Q(-12, 4) using θ = arctan ( x y ) and adjust for the quadrant, resulting in θ ≈ 161. 6 ∘ .
P ( 6 , 3 ) ≈ ( 6.7 , 26. 6 ∘ ) , Q ( − 12 , 4 ) ≈ ( 12.6 , 161. 6 ∘ )
Explanation
Problem Analysis The problem asks us to convert Cartesian coordinates to polar coordinates. We are given two points, P(6, 3) and Q(-12, 4), and we need to find their corresponding polar coordinates (r, θ), where r is the radial distance from the origin and θ is the angle in degrees, measured counterclockwise from the positive x-axis. We are instructed not to use the polar conversion function on our calculator and to express the angle to one decimal place.
Convert P(6,3) to polar coordinates For point P(6, 3), we calculate the radial distance r using the formula: r = x 2 + y 2 Substituting the coordinates of P, we get: r = 6 2 + 3 2 = 36 + 9 = 45 ≈ 6.7 Next, we calculate the angle θ using the formula: θ = arctan ( x y ) Substituting the coordinates of P, we get: θ = arctan ( 6 3 ) = arctan ( 0.5 ) ≈ 26. 6 ∘ Since P(6, 3) is in the first quadrant, the angle is simply 26.6°.
Convert Q(-12,4) to polar coordinates For point Q(-12, 4), we calculate the radial distance r using the formula: r = x 2 + y 2 Substituting the coordinates of Q, we get: r = ( − 12 ) 2 + 4 2 = 144 + 16 = 160 ≈ 12.6 Next, we calculate the angle θ using the formula: θ = arctan ( x y ) Substituting the coordinates of Q, we get: θ = arctan ( − 12 4 ) = arctan ( − 3 1 ) ≈ − 18. 4 ∘ Since Q(-12, 4) is in the second quadrant, we need to add 180° to the result to get the correct angle: θ = − 18. 4 ∘ + 18 0 ∘ = 161. 6 ∘
Final Answer Therefore, the polar coordinates for P(6, 3) are approximately (6.7, 26.6°) and for Q(-12, 4) are approximately (12.6, 161.6°).
Examples
Polar coordinates are useful in navigation and mapping. For example, an air traffic controller might use polar coordinates to track the position of an airplane relative to the airport. The radial distance tells how far away the plane is, and the angle tells the direction from the airport. This makes it easy to communicate the plane's location to other controllers or pilots.
