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

Problem

(5,(5*π)/3)

Solution

1. Identify the given coordinates. The problem provides a point in polar coordinates (r,θ) where r=5 and θ=(5*π)/3 Since the task asks to convert to polar coordinates but the input is already in polar form, we will convert these polar coordinates to rectangular coordinates (x,y)

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

x=5*cos((5*π)/3)

3. Evaluate the cosine function. The angle (5*π)/3 is in the fourth quadrant, where cosine is positive. The reference angle is π/3

cos((5*π)/3)=1/2

4. Calculate the value of x

x=5⋅1/2=5/2

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

y=5*sin((5*π)/3)

6. Evaluate the sine function. In the fourth quadrant, sine is negative.

sin((5*π)/3)=−√(,3)/2

7. Calculate the value of y

y=5⋅(−√(,3)/2)=−(5√(,3))/2

Final Answer

(5,(5*π)/3)=(5/2,−(5√(,3))/2)

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