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The explanation for why we use 1/cos(theta) does not make any sense, especially given what was taught in the worksheet videos :(.

Hi there! Cut the coil in the middle at 45 degrees and join the circular ends. We have created elliptical cross-section. The axis of ellipse and axis of circular coil are the same, the usual 45 degrees still applies.

Use a cuboid and trigonometry, we can prove the area increases by a factor 1/ (cos 45 degrees).

Hi there! If we chop the coil so that its cross-sectional area is parallel to the loop's area, the cross-sectional area is oval/elliptical. The elliptical area is greater than the circular area, this is why we divide by cos(45).

The axis of the area and axis of loop(B), make a 45 degree angle, so we multiply by the usual cos(45). That is, even though the cross-sectional area becomes elliptical, the axis maintains its position as it were in the circle.

Intuitively: Take two cylinders/pipes of equal length. Keep one as it is.

Cut the other pipe anywhere in the middle at an angle of 45 degrees, join its circular ends. The cylinder has elliptical cross-sections at the ends. The length of the cylinder remains the same at all points - meaning we created many identical elliptical areas along the pipe, consistent with the idea of cross-sectional area.

Explaining the increase in area using a cuboid: Cut a cuboid at 45 degrees and join the perpendicular sides, leaving sloping faces at the ends. The length and volume remain unchanged. The cross-sectional area increases since one of the sides used to compute cross-sectional area is replaced with hypotenuse. Using trigonometry and image below, cos (Q) = adj/hyp = h/c. Solving for the hypotence, c = h/cos(Q). The cross-section was initially (width × h), now it has changed to (width × c) = wh/cosQ. If we used angle P, we would end up with wh/sin(P). Thus, choice of angle matters.

More reasoning/logic: If we join the elliptical ends, we can still re-construct out circular cylinder. I am simply saying that a pipe with circular ends and a pipe with elliptical ends is the same thing as long as it is directed at 45 degrees relative to the square loop. The volume of the coil remains the same, but the cross-sectional are changes.