Convert the point to rectangular coordinates.(4,π6)(4,\frac{\pi}{6})(4,6π)

( 2 3 , 2 ) (2\sqrt{3},2) ( 2 3 <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"> , 2 )

Convert the point to rectangular coordinates. ( 4 , π 6 ) (4,\frac{\pi}{6}) ( 4 , 6 π )

Here we're asked to convert the point given in polar coordinates into rectangular coordinates. Now the point that we're given here is four pi over six. So let's go ahead and use our formulas here in order to get our x and y values. Now x is going to be equal to r times the cosine of theta and here our r value is four, our theta value is pi over six, so this is going to give me four times the cosine of pi over six. Now the cosine of pi over six is root three over two, so this is going to be four times the square root of three over two. Now doing some simplification here, four over two just gives me two, so this simplifies to two root three for that x value. Now for y, y is equal to r times the sine of theta. So plugging it in r and theta again here we get four times the sine of pi over six. Now the sine of pi over six is one half so this gives me four times one half. Now four times one half is just two, so that gives me my final y value and this point in rectangular coordinates is therefore two root three comma two. Now this is our final answer here, having converted our point from polar coordinates into rectangular coordinates. Thanks for watching, and I'll see you in the next one.

Convert the point to rectangular coordinates. ( 4 , π 6 ) (4,\frac{\pi}{6}) ( 4 , 6 π )

( 2 3 , 2 ) (2\sqrt{3},2) ( 2 3 <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"> , 2 )

( 4 3 , 4 ) (4\sqrt{3},4) ( 4 3 <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 )

( 2 , 2 3 ) (2,2\sqrt{3}) ( 2 , 2 3 <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"> )

( 2 , 3 ) (2,\sqrt{3}) ( 2 , 3 <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"> )

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

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Multiple Choice

Convert the point to rectangular coordinates.
$(4,\frac{\pi}{6})$

$(2\sqrt{3},2)$

$(4\sqrt{3},4)$

$(2,2\sqrt{3})$

$(2,\sqrt{3})$

Verified step by step guidance

Step 1: Understand the problem. The given point is in polar coordinates (r, θ), where r is the radius (distance from the origin) and θ is the angle in radians. We need to convert this point to rectangular coordinates (x, y).

Step 2: Recall the formulas for converting polar coordinates to rectangular coordinates: x = r * cos(θ) and y = r * sin(θ).

Step 3: Substitute the given values into the formulas. Here, r = 4 and θ = π/6. So, x = 4 * cos(π/6) and y = 4 * sin(π/6).

Step 4: Use the trigonometric values for cos(π/6) and sin(π/6). From the unit circle, cos(π/6) = √3/2 and sin(π/6) = 1/2.

Step 5: Simplify the expressions for x and y. For x, calculate 4 * (√3/2), and for y, calculate 4 * (1/2). This will give the rectangular coordinates (x, y).

5:32m

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