Convert the point to rectangular coordinates.(−2,−π4)(-2,-\frac{\pi}{4})(−2,−4π)

(−2,2)(-\sqrt{2},\sqrt{2})(−2,2)

Convert the point to rectangular coordinates. ( − 2 , − π 4 ) (-2,-\frac{\pi}{4}) ( − 2 , − 4 π )

Here, we're asked to convert the point given in polar coordinates into rectangular coordinates, and the point that we're given is a negative 2, negative pi over 4. So let's go ahead and use our formulas here in order to get our x and y values to get our point in rectangular coordinates. Now x is going to be equal to r times the cosine of theta, and here our r value is negative 2 while our theta value is negative pi over 4. So x is going to be equal to negative 2 times the cosine of negative pi over 4. Now the cosine of negative pi over 4 is root 2 over 2 so this will be negative 2 times a root 2 over 2. Now simplifying this here allows these 2's to cancel out and we are just left with negative root 2 for our x value. Then for y, we know that y is equal to r times the sine of beta. So again, plugging in our r and theta values, we get negative 2 times the sine of negative pi over 4. Now the sine of negative pi over 4 is negative root 2 over 2, so this will end up giving me a negative 2 times a negative root 2 over 2. Now here canceling some stuff out I'm able to cancel these twos and also these negatives because 2 negatives makes a positive. This just leaves me with the square root of 2 for that y value. So this point in polar coordinates will be negative root 2 comma root 2, and this is our final answer here. Having converted our original polar coordinates into their rectangular coordinates. Thanks for watching, and let me know if you have questions.

Convert the point to rectangular coordinates.(−2,−π4)(-2,-\frac{\pi}{4})(−2,−4π)

(−2,2)(-\sqrt{2},\sqrt{2})(−2,2)

(2,−2)(\sqrt{2},-\sqrt2)(2,−2)

(−22,22)(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})(−22,22)

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15. Polar Equations

Convert Points Between Polar and Rectangular Coordinates

Precalculus

15. Polar Equations

Convert Points Between Polar and Rectangular Coordinates

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Multiple Choice

Convert the point to rectangular coordinates.
$(-2,-$\frac{\pi}{4}$)$

$($\sqrt{2}$,-$\sqrt$2)$

$(-1,1)$

$(-$\sqrt{2}$,$\sqrt{2}$)$

$(-$\frac{\sqrt{2}$}{2},$\frac{\sqrt{2}$}{2})$

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Verified step by step guidance

Understand that the given point is in polar coordinates, which are represented as (r, θ), where r is the radius and θ is the angle.

To convert polar coordinates to rectangular coordinates (x, y), use the formulas: x = r * cos(θ) and y = r * sin(θ).

For the point (-2, -π/4), substitute r = -2 and θ = -π/4 into the formulas: x = -2 * cos(-π/4) and y = -2 * sin(-π/4).

Calculate cos(-π/4) and sin(-π/4). Remember that cos(-θ) = cos(θ) and sin(-θ) = -sin(θ). Therefore, cos(-π/4) = cos(π/4) = √2/2 and sin(-π/4) = -sin(π/4) = -√2/2.

Substitute these values into the equations: x = -2 * (√2/2) and y = -2 * (-√2/2), and simplify to find the rectangular coordinates.