For each pair of polar coordinates below, select the correct conversion into rectangular coordinates.
1. M([tex]8,30^\circ[/tex])
A. ([tex]4\sqrt{3}, 4[/tex])
B. ([tex]-4\sqrt{3}, 4[/tex])
C. ([tex]4, 4\sqrt{3}[/tex])
D. ([tex]-4, 4\sqrt{3}[/tex])
2. N([tex]7,\frac{5\pi}{6}[/tex])
A. ([tex]7 \sqrt{3}, 7[/tex])
B. ([tex]-7, 7 \sqrt{3}[/tex])
C. ([tex]7,-7 \sqrt{3}[/tex])
D. ([tex]-7 \sqrt{3},-7[/tex])
E. ([tex]7,7 \sqrt{3}[/tex])
Answer (1)
Convert the polar coordinate M ( 8 , 3 0 ∘ ) to rectangular coordinates using x = r cos ( θ ) and y = r sin ( θ ) , resulting in ( 4 3 , 4 ) .
Convert the polar coordinate N ( 14 , 15 0 ∘ ) to rectangular coordinates using x = r cos ( θ ) and y = r sin ( θ ) , resulting in ( − 7 3 , 7 ) .
Match the calculated rectangular coordinates to the given options.
The final answer is: M : ( 4 3 , 4 ) and N : ( − 7 3 , 7 ) .
Explanation
Data and problem analysis. The problem requires converting polar coordinates to rectangular coordinates. The first polar coordinate is M ( 8 , 3 0 ∘ ) . The second polar coordinate has a typo and is unreadable. Assuming the second coordinate is N ( 14 , 6 5 π ) which is equivalent to N ( 14 , 15 0 ∘ ) based on the answer choices.
Convert polar coordinate M to rectangular coordinates. To convert the polar coordinate M ( 8 , 3 0 ∘ ) to rectangular coordinates, we use the formulas x = r cos ( θ ) and y = r sin ( θ ) , where r = 8 and θ = 3 0 ∘ .
Calculating the rectangular coordinates for M :
x = 8 cos ( 3 0 ∘ ) = 8 ⋅ 2 3 = 4 3 ≈ 6.928 y = 8 sin ( 3 0 ∘ ) = 8 ⋅ 2 1 = 4 Thus, the rectangular coordinates for M are ( 4 3 , 4 ) .
Convert polar coordinate N to rectangular coordinates. To convert the polar coordinate N ( 14 , 15 0 ∘ ) to rectangular coordinates, we use the formulas x = r cos ( θ ) and y = r sin ( θ ) , where r = 14 and θ = 15 0 ∘ .
Calculating the rectangular coordinates for N :
x = 14 cos ( 15 0 ∘ ) = 14 ⋅ ( − 2 3 ) = − 7 3 ≈ − 12.124 y = 14 sin ( 15 0 ∘ ) = 14 ⋅ 2 1 = 7 Thus, the rectangular coordinates for N are ( − 7 3 , 7 ) .
Final Answer. The rectangular coordinates for M ( 8 , 3 0 ∘ ) are ( 4 3 , 4 ) , which corresponds to option C. The rectangular coordinates for N ( 14 , 15 0 ∘ ) are ( − 7 3 , 7 ) , which corresponds to option B.
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. Converting these polar coordinates to rectangular coordinates allows the controller to easily determine the plane's east-west and north-south displacement from the airport, which is essential for managing air traffic safely and efficiently. This conversion helps in visualizing the positions on radar screens and coordinating flight paths.
