Convert to Polar Coordinates (2,(7pi)/4)

Problem

(2,(7*π)/4)

Solution

1. Identify the given coordinates as polar coordinates (r,θ) where r=2 and θ=(7*π)/4

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

x=2*cos((7*π)/4)

3. Evaluate the cosine of (7*π)/4 which is in the fourth quadrant where cosine is positive.

cos((7*π)/4)=√(,2)/2

4. Calculate the x-value by multiplying the radius by the cosine value.

x=2⋅√(,2)/2=√(,2)

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

y=2*sin((7*π)/4)

6. Evaluate the sine of (7*π)/4 which is in the fourth quadrant where sine is negative.

sin((7*π)/4)=−√(,2)/2

7. Calculate the y-value by multiplying the radius by the sine value.

y=2⋅(−√(,2)/2)=−√(,2)

Final Answer

(2,(7*π)/4)=(√(,2),−√(,2))

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