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

Problem

(r,θ)=(−3,(5*π)/3)

Solution

1. Identify the polar coordinates r and θ from the given point.

r=−3

θ=(5*π)/3

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

x=−3*cos((5*π)/3)

3. Evaluate the cosine of (5*π)/3 Since (5*π)/3 is in the fourth quadrant, the cosine is positive.

cos((5*π)/3)=1/2

4. Calculate the value of x

x=−3⋅1/2

x=−3/2

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

y=−3*sin((5*π)/3)

6. Evaluate the sine of (5*π)/3 Since (5*π)/3 is in the fourth quadrant, the sine is negative.

sin((5*π)/3)=−√(,3)/2

7. Calculate the value of y

y=−3⋅(−√(,3)/2)

y=(3√(,3))/2

Final Answer

(−3,(5*π)/3)=(−3/2,(3√(,3))/2)

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