Find the distance between P1(1,3,-47°) and P2(-3.6,-62°).

Steps to Solve

1. Identify Coordinates

   We have the following polar coordinates:

   • ( P_1 = (1.3, -47^\circ) )
   • ( P_2 = (-3.6, -62^\circ) )

2. Convert Polar to Cartesian Coordinates

   Use the formulas ( x = r \cos(\theta) ) and ( y = r \sin(\theta) ) to convert the polar coordinates to Cartesian coordinates.

   For ( P_1 ): [ x_1 = 1.3 \cos(-47^\circ) ] [ y_1 = 1.3 \sin(-47^\circ) ]

   For ( P_2 ): [ x_2 = -3.6 \cos(-62^\circ) ] [ y_2 = -3.6 \sin(-62^\circ) ]

3. Calculate ( x_1 ) and ( y_1 )

   Substitute ( \theta = -47^\circ ) into the equations: [ x_1 = 1.3 \cos(-47^\circ) \approx 1.3 \cdot 0.68199 \approx 0.887 ] [ y_1 = 1.3 \sin(-47^\circ) \approx 1.3 \cdot -0.73135 \approx -0.951 ]

4. Calculate ( x_2 ) and ( y_2 )

   Substitute ( \theta = -62^\circ ) into the equations: [ x_2 = -3.6 \cos(-62^\circ) \approx -3.6 \cdot 0.46946 \approx -1.688 ] [ y_2 = -3.6 \sin(-62^\circ) \approx -3.6 \cdot -0.88295 \approx 3.182 ]

5. Apply the Distance Formula

   Use the distance formula: [ P_1P_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] Substitute ( x_1, y_1, x_2, y_2 ): [ P_1P_2 = \sqrt{((-1.688 - 0.887)^2 + (3.182 - (-0.951))^2)} ]

6. Calculate the Distance

   Simplify and calculate: [ P_1P_2 = \sqrt{(-2.575)^2 + (4.133)^2} \approx \sqrt{6.639 + 17.070} \approx \sqrt{23.709} ]

   Therefore, [ P_1P_2 \approx 4.87 ]