Find the distance between P1(3, –195°) and P2(–4, –94°) on the polar plane. Round your answer to the nearest thousandth.
P2(-4,-94°) can also be written a (4,94°)
Make a sketch to see that you have two straight lines with vertex at the origin of lenghts 3 and 4 with an angle of 79° between them.
by the cosine law:
(P1P2)^2 = 3^2 + 4^2 - 2(3)(4)cos 79°
= 20.42056
P1P2 = 4.519
Thanks, I am understanding now.
To find the distance between two points on the polar plane, you can use the formula:
distance = √[ (r2^2) + (r1^2) - 2(r2)(r1)cos(θ2 - θ1) ]
where r1 and θ1 are the polar coordinates of the first point, and r2 and θ2 are the polar coordinates of the second point.
Given that P1 has coordinates (3, -195°) and P2 has coordinates (-4, -94°), we can substitute the values into the formula:
distance = √[ (-4^2) + (3^2) - 2(-4)(3)cos(-94° - (-195°)) ]
Simplifying further:
distance = √[ 16 + 9 + 24cos(-94° + 195°) ]
distance = √[ 25 + 24cos(101°) ]
Now, calculate the value of cos(101°) using a calculator.
cos(101°) ≈ 0.34202
Substituting the value back into the equation:
distance = √[ 25 + 24 * 0.34202 ]
distance = √[ 25 + 8.20848 ]
distance = √33.20848
Finally, round the answer to the nearest thousandth to get:
distance ≈ 5.765