The polar coordinates of a point P are (2,240 degrees). The Cartesian (rectangular) coordinates of P are
(A,B), where A = 2 * cos(240 degrees) and B = 2 * sin(240 degrees).
We know that 240 degrees is equal to 180 degrees + 60 degrees. So we can write the equations as:
A = 2 * cos(180 + 60) = 2 * cos(180) * cos(60) - 2 * sin(180) * sin(60)
B = 2 * sin(180 + 60) = 2 * sin(180) * cos(60) + 2 * cos(180) * sin(60)
Now we know that:
cos(60) = 1/2, sin(60) = √3/2
cos(180) = -1, sin(180) = 0
Using this information, we can calculate A and B:
A = 2 * (-1) * (1/2) - 2 * 0 * (√3/2) = -2 * (1/2) = -1
B = 2 * 0 * (1/2) + 2 * (-1) * (√3/2) = 2 * (-1) * (√3/2) = -√3
So the Cartesian coordinates of P are (-1, -√3).
To convert the polar coordinates (r, θ) to Cartesian coordinates (x, y), we can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
In this case, the given polar coordinates are (2, 240 degrees).
Using the formulas, we can substitute the values:
x = 2 * cos(240 degrees)
y = 2 * sin(240 degrees)
But before we can calculate the exact values of x and y, we have to convert the angle from degrees to radians, because most programming languages use radians for trigonometric calculations.
To convert the angle from degrees to radians, we can use the following formula:
radians = degrees * π / 180
Substituting the given angle of 240 degrees:
radians = 240 * π / 180
Now, we can calculate the values of x and y:
x = 2 * cos(240 * π / 180)
y = 2 * sin(240 * π / 180)
Evaluating the trigonometric functions using radians, we get:
x ≈ 2 * cos(4.18879)
y ≈ 2 * sin(4.18879)
Now, you can use a calculator or a programming language that supports trigonometric calculations to find the approximate values of x and y.
To find the Cartesian coordinates of a point given its polar coordinates, you can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
In this case, the polar coordinates of point P are (2, 240 degrees). Plugging these values into the formulas:
x = 2 * cos(240 degrees)
y = 2 * sin(240 degrees)
To find the values of cos(240 degrees) and sin(240 degrees), we need to convert the angle from degrees to radians. Here's how:
1. Convert 240 degrees to radians:
theta_rad = 240 degrees * (pi/180 degrees)
2. Evaluate cos(theta_rad) and sin(theta_rad) using a calculator:
cos(240 degrees) = cos(240 degrees * (pi/180 degrees))
sin(240 degrees) = sin(240 degrees * (pi/180 degrees))
After evaluating these trigonometric functions, you can substitute the values back into the equations for x and y:
x = 2 * cos(240 degrees)
y = 2 * sin(240 degrees)
These will give you the Cartesian coordinates (x, y) of point P.