consider the polar coordinates (3,4pi/3). Find the equivalent rectangular coordinate.
To find the equivalent rectangular coordinate from the given polar coordinates, we can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
Here, r represents the magnitude or distance from the origin, and theta represents the angle measured counterclockwise from the positive x-axis.
For the given polar coordinates (3, 4π/3), the magnitude or distance from the origin is 3, and the angle is 4π/3.
Using the formulas, we can calculate:
x = 3 * cos(4π/3)
y = 3 * sin(4π/3)
Now let's break down the calculations step by step.
Step 1:
Plug in the values into the formulas:
x = 3 * cos(4π/3)
y = 3 * sin(4π/3)
Step 2:
Evaluate cos(4π/3) using the unit circle or trigonometric identities:
cos(4π/3) = -1/2
Step 3:
Evaluate sin(4π/3) using the unit circle or trigonometric identities:
sin(4π/3) = -√3/2
Step 4:
Calculate x and y by substituting the values:
x = 3 * (-1/2) = -3/2
y = 3 * (-√3/2) = -3√3/2
Therefore, the equivalent rectangular coordinate for the given polar coordinates (3, 4π/3) is (-3/2, -3√3/2).
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