Convert the rectangular coordinate (2,-2) to polar form and find two additional representations of this point.

Can you check my work and explain how to finish the question?

r=√(2)2+(-2)2
r=2√2

tan theta = (-2)/2
theta = -1

polar coordinate= (2√2, -1)

no,

tanØ = -1 , not Ø = -1

since (2,-2) is in quadrant IV, Ø = 315° or 7π/4

so one answer is (2√2 , 7π/4)
another would be (-2√2, 3π/4)

Well, it seems like you made a little mistake in finding the angle, theta. Let me help you out.

To convert the rectangular coordinate (2, -2) to polar form, we need to find the magnitude (r) and the angle (theta).

To find the magnitude (r), we use the formula:
r = √(x^2 + y^2)
where x and y are the coordinates of the point.
r = √(2^2 + (-2)^2)
r = √(4 + 4)
r = √8 = 2√2

Now, to find the angle (theta), we use the formula:
tan(theta) = y/x
theta = atan(y/x)
where atan represents the arctan function.

theta = atan(-2/2)
theta = atan(-1)

Note that atan only gives the angle between -π/2 and π/2, so we have to determine the exact angle based on the provided rectangular coordinates. In this case, since the x-coordinate is positive and the y-coordinate is negative, we know that the point lies in the fourth quadrant, so the angle will be in the range of -π/2 to -π.

Therefore, theta = -π + atan(-1)
theta = -1.57 - 1
theta = -2.57 (approximately)

So, the polar form of the rectangular coordinate (2, -2) is (2√2, -2.57).

Now, let's find two additional representations of this point. Remember that we can add or subtract any multiple of 2π to the angle (θ) while keeping the magnitude (r) constant.

1. (2√2, -2.57 + 2π) - Adding 2π to the angle:
This will give us another representation of the point in the same position but with a different angle measurement. Remember to adjust the angle to be within the range of -π to π.
(-2√2, 3.48) - This is one representation.

2. (2√2, -2.57 - 2π) - Subtracting 2π from the angle:
This will give us another representation of the point in the same position but with a different angle measurement. Again, remember to adjust the angle to be within the range of -π to π.
(-2√2, -5.70) - This is the second representation.

I hope this clarifies things for you!

To convert the rectangular coordinate (2,-2) to polar form, follow these steps:

1. Calculate the value of r, the distance from the origin to the point using the formula: r = √(x^2 + y^2), where x and y are the rectangular coordinates.

r = √(2^2 + (-2)^2)
r = √(4 + 4)
r = √8 = 2√2

2. To find the value of theta (angle), you can use the tan^-1 function or inverse tangent. Calculate the tangent of theta using the formula: tan(theta) = y/x.

tan(theta) = (-2)/2
theta = tan^-1(-1)
theta ≈ -45 degrees

3. Now, you can represent the point in polar form as (r, theta): (2√2, -45 degrees).

To find two additional representations of this point, we can add or subtract multiples of 360 degrees from the angle. This is because the angle is periodic, and adding or subtracting 360 degrees will result in the same point on the polar coordinate system.

4. Add 360 degrees to the angle:

theta + 360 degrees ≈ -45 degrees + 360 degrees
theta ≈ 315 degrees

So, another representation of the point in polar form is (2√2, 315 degrees).

5. Subtract 360 degrees from the angle:

theta - 360 degrees ≈ -45 degrees - 360 degrees
theta ≈ -405 degrees

However, -405 degrees can be represented as -405 degrees + 360 degrees = -45 degrees.

Therefore, the final representation of the point in polar form is (2√2, -45 degrees), (2√2, 315 degrees), and (2√2, -45 degrees).

To convert the rectangular coordinate (2, -2) to polar form, you need to find the magnitude (r) and angle (θ) of the point.

First, find the magnitude (r) using the formula:
r = √(x^2 + y^2)
where x and y are the coordinates of the point.

In this case, substitute x = 2 and y = -2 into the formula:
r = √(2^2 + (-2)^2)
r = √(4 + 4)
r = √8
r = 2√2

Next, find the angle (θ) using the formula:
θ = tan^(-1)(y/x)
where x and y are the coordinates of the point.

In this case, substitute x = 2 and y = -2 into the formula:
θ = tan^(-1)(-2/2)
θ = tan^(-1)(-1)
θ = -1 (in radians)

Therefore, the polar form of the rectangular coordinate (2, -2) is (2√2, -1).

To find two additional representations of this point, you can add or subtract multiples of 2π (360 degrees) to the angle (θ). This is because in polar form, multiple representations of the same point are possible due to the periodic nature of angles.

For example, adding 2π to the angle (-1) gives:
θ' = -1 + 2π = 5.28 (in radians)

Substituting the new angle into the polar form equation, we get:
(2√2, 5.28)

Similarly, subtracting 2π from the angle (-1) gives:
θ'' = -1 - 2π = -7.28 (in radians)

Substituting the new angle into the polar form equation, we get:
(2√2, -7.28)

So, the two additional representations of the point (2, -2) in polar form are (2√2, 5.28) and (2√2, -7.28).