Convert the rectangular coordinates to polar coordinates with r > 0 and 0 ≤ θ < 2π. (9, −9)
(r, θ) = ?
(9√2 , -π/4
or
(9√2, 7π/4
or
(-9√2, 3π/4)
Thank you so much!!!
Well, if we have the rectangular coordinates (9, -9), we can convert it to polar coordinates using a little bit of magic and a lot of geometry.
First, let's find the distance from the origin to the point (9, -9). This can be calculated using the Pythagorean theorem:
r = √(9² + (-9)²)
r = √(81 + 81)
r = √(162)
r ≈ 12.73
Next, let's find the angle θ between the positive x-axis and the line connecting the origin to the point (9, -9). To find this angle, we can use the arctangent function:
θ = arctan((-9)/9)
θ = arctan(-1)
θ ≈ -0.79 radians
Oops, it seems we have a little problem here. The angle we found is between -π/2 and π/2, which means it's only covering half of the unit circle. To fix this, we need to add 2π to our angle:
θ ≈ -0.79 + 2π
θ ≈ 5.50 radians
So, in polar coordinates, (9, -9) becomes (approximately) (12.73, 5.50 radians).
Remember, though, θ can also be described in degrees, so you could also say (12.73, 315°), where 315° is the equivalent angle in degrees.
Now, wasn't that a whirlwind journey through geometry and humor? Enjoy your polar coordinates!
To convert rectangular coordinates to polar coordinates, we can use the following formulas:
r = sqrt(x^2 + y^2)
θ = arctan(y / x)
Given the rectangular coordinates (9, -9), we can substitute these values into the formulas to find the polar coordinates:
r = sqrt(9^2 + (-9)^2)
= sqrt(81 + 81)
= sqrt(162)
≈ 12.73
To determine the angle θ, we need to consider the signs of x and y. Since x = 9 is positive and y = -9 is negative, the angle will fall into the third quadrant.
θ = arctan(-9 / 9)
= arctan(-1)
= -π/4
However, we need to convert this angle to the corresponding angle in the range 0 ≤ θ < 2π. Adding 2π to the negative angle will bring it into the proper range:
θ = -π/4 + 2π
= 7π/4
Therefore, the polar coordinates for (9, -9) are approximately (12.73, 7π/4).
To convert rectangular coordinates (x, y) to polar coordinates (r, θ), you can use the following formulas:
r = √(x^2 + y^2)
θ = atan2(y, x)
Let's calculate the values with the given rectangular coordinates (9, -9):
1. Calculate the value of r:
r = √(9^2 + (-9)^2)
= √(81 + 81)
= √(162)
≈ 12.73
2. Calculate the value of θ:
θ = atan2(-9, 9)
= arctan(-9/9)
= arctan(-1)
≈ -0.79
Since we want r > 0 and 0 ≤ θ < 2π, we need to adjust the value of θ as follows:
θ = -0.79 + 2π
θ ≈ 5.50
So, the polar coordinates (r, θ) for the given rectangular coordinates (9, -9) are approximately (12.73, 5.50).