Convert the point to polar coordinates.(1,1)(1,1)(1,1)

( 2 , π 4 ) (\sqrt{2},\frac{\pi}{4}) ( 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , 4 π )

Convert the point to polar coordinates. ( 1 , 1 ) (1,1) ( 1 , 1 )

In this problem, what we're asked to convert the point given in rectangular coordinates into polar coordinates, and the point that we're given here is one one. So let's start with step one and go ahead and plot our point. Now plotting our point, I end up right here in quadrant one. Now moving on to step two, calculating our r value, this gives me the square root of one squared plus one squared, which will end up being the square root of two for that r value. Now moving on to finding theta, here we look at the location of our point. Now here it's located in quadrant one, which tells me that in order to find theta, all I need to do is take the inverse tangent of y over x. Now plugging my y and x values in there, I end up getting that theta is equal to the inverse tangent of one over one, which we know is just one. So this is the inverse tangent of one and if I type this into my calculator or I use my unit circle, I will see that this is equal to pi over four for that beta value. Now I have my point in polar coordinates at root two pi over four, and we're done here with our final answer. Thanks for watching, and I'll see you in the next one.

Convert the point to polar coordinates. ( 1 , 1 ) (1,1) ( 1 , 1 )

( 2 , π 4 ) (\sqrt{2},\frac{\pi}{4}) ( 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , 4 π )

( 1 , π 4 ) (1,\frac{\pi}{4}) ( 1 , 4 π )

( 2 , 5 π 4 ) (\sqrt{2},\frac{5\pi}{4}) ( 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , 4 5 π )

( 2 , − π 4 ) (\sqrt{2},-\frac{\pi}{4}) ( 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , − 4 π )

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Struggling with Calculus?

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Multiple Choice

Convert the point to polar coordinates.
$(1,1)$

$(1,\frac{\pi}{4})$

$(\sqrt{2},\frac{\pi}{4})$

$(\sqrt{2},\frac{5\pi}{4})$

$(\sqrt{2},-\frac{\pi}{4})$

Verified step by step guidance

Step 1: Recall the formula for converting Cartesian coordinates (x, y) to polar coordinates (r, θ). The radius r is given by r = √(x² + y²), and the angle θ is given by θ = arctan(y/x).

Step 2: Substitute the given Cartesian coordinates (1, 1) into the formula for r. Compute r = √(1² + 1²) = √2.

Step 3: Substitute the given Cartesian coordinates (1, 1) into the formula for θ. Compute θ = arctan(1/1) = arctan(1). Recall that arctan(1) corresponds to π/4 radians in the first quadrant.

Step 4: Combine the results to express the polar coordinates. The polar coordinates are (r, θ) = (√2, π/4).

Step 5: Verify the solution by considering alternative representations of the angle θ. For example, θ could also be expressed as −π/4 or 5π/4 depending on the quadrant, but the most common representation for the point (1, 1) is (√2, π/4).

5:32m

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