If the rectangular coordinates of a point are given by (4,y), and its polar coordinates are (r,25 degrees), determine y and r.
To determine the values of y and r, we can use the formulas for converting between rectangular and polar coordinates.
Rectangular to Polar Conversion:
Given the rectangular coordinates (x, y), the polar coordinates (r, θ) can be found using the following formulas:
r = √(x^2 + y^2) -- equation (1)
θ = arctan(y/x) -- equation (2)
Polar to Rectangular Conversion:
Given the polar coordinates (r, θ), the rectangular coordinates (x, y) can be found using the following formulas:
x = r * cos(θ)
y = r * sin(θ)
Now let's use these formulas to find the values of y and r.
Given:
Rectangular coordinates: (4, y)
Polar coordinates: (r, 25 degrees)
1. To find y:
Since the x-coordinate is given as 4, we'll use equation (2) to find y.
θ = 25 degrees
x = 4
Using equation (2):
y = x * tan(θ)
y = 4 * tan(25 degrees)
Calculating the value of y:
y ≈ 1.747
2. To find r:
To find r, we'll use equation (1) with the given rectangular coordinates (4, y).
x = 4
y = 1.747
Using equation (1):
r = √(x^2 + y^2)
r = √(4^2 + 1.747^2)
Calculating the value of r:
r ≈ 4.440
Therefore, the values of y and r are approximately:
y ≈ 1.747
r ≈ 4.440