Convert to Rectangular Coordinates (-3,(7pi)/6)

Problem

(r,θ)=(−3,(7*π)/6)

Solution

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

r=−3

θ=(7*π)/6

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

x=−3*cos((7*π)/6)

3. Evaluate the cosine of (7*π)/6 which is in the third quadrant.

cos((7*π)/6)=−√(,3)/2

4. Calculate the value of x by multiplying.

x=−3*(−√(,3)/2)

x=(3√(,3))/2

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

y=−3*sin((7*π)/6)

6. Evaluate the sine of (7*π)/6

sin((7*π)/6)=−1/2

7. Calculate the value of y by multiplying.

y=−3*(−1/2)

y=3/2

Final Answer

(r,θ)=(−3,(7*π)/6)⇒(x,y)=((3√(,3))/2,3/2)

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