Find the rectangular coordinates of $(5,30^{\circ})$.
A. $(\frac{5 \sqrt{3}}{2},-\frac{5}{2})$
B. $(-\frac{5 \sqrt{3}}{2},-\frac{5}{2})$
C. $(\frac{5 \sqrt{3}}{2}, \frac{5}{2})$
D. $(-\frac{5 \sqrt{3}}{2}, \frac{5}{2})$
Answer (2)
Convert polar coordinates to rectangular coordinates.
Use the formulas x = r cos ( θ ) and y = r sin ( θ ) .
Substitute r = 5 and θ = 3 0 ∘ into the formulas.
The rectangular coordinates are ( 2 5 3 , 2 5 ) .
Explanation
Problem Setup We are given the polar coordinates ( 5 , 3 0 ∘ ) and we want to find the corresponding rectangular coordinates ( x , y ) .
Conversion Formulas To convert from polar coordinates ( r , θ ) to rectangular coordinates ( x , y ) , we use the following formulas:
x = r cos ( θ ) y = r sin ( θ )
Values of Trigonometric Functions In this case, r = 5 and θ = 3 0 ∘ . We know that cos ( 3 0 ∘ ) = 2 3 and sin ( 3 0 ∘ ) = 2 1 .
Calculate x and y Substituting these values into the conversion formulas, we get:
x = 5 cos ( 3 0 ∘ ) = 5 ⋅ 2 3 = 2 5 3 y = 5 sin ( 3 0 ∘ ) = 5 ⋅ 2 1 = 2 5
Final Answer Therefore, the rectangular coordinates are ( 2 5 3 , 2 5 ) .
Examples
Polar coordinates are useful in navigation, especially in aviation and maritime contexts. For example, an air traffic controller might use polar coordinates to track an airplane's position relative to the airport. If the radar shows a plane at a distance of 5 miles and an angle of 30 degrees from the east, converting these polar coordinates to rectangular coordinates helps determine the plane's exact location on a grid, which is essential for guiding the aircraft safely. This conversion allows for precise positioning and collision avoidance.
To convert polar coordinates ( 5 , 3 0 ∘ ) to rectangular coordinates, we use the formulas x = r cos ( θ ) and y = r sin ( θ ) . By substituting the values, we find that the rectangular coordinates are ( 2 5 3 , 2 5 ) . Therefore, the correct choice is C.
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