Convert to Rectangular Coordinates (3,(5pi)/3)

Problem

(3,(5*π)/3)

Solution

1. Identify the polar coordinates (r,θ) from the given point, where r=3 and θ=(5*π)/3

2. Apply the formula for the x-coordinate using x=r*cos(θ)

3. Substitute the values into the x-coordinate formula:

x=3*cos((5*π)/3)

4. Evaluate the cosine function, noting that (5*π)/3 is in the fourth quadrant where cosine is positive:

x=3*(1/2)=3/2

5. Apply the formula for the y-coordinate using y=r*sin(θ)

6. Substitute the values into the y-coordinate formula:

y=3*sin((5*π)/3)

7. Evaluate the sine function, noting that (5*π)/3 is in the fourth quadrant where sine is negative:

y=3*(−√(,3)/2)=−(3√(,3))/2

Final Answer

(3,(5*π)/3)=(3/2,−(3√(,3))/2)

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