Trigonometry in 3D and complex figures

A wedge-shaped door stop has a right-angled triangle at each end: four centimetres along the floor and three centimetres up. It runs back twelve centimetres. How long is a straight line from the sharp bottom corner at the near end to the top corner at the far end? It isn't a box, and no triangle is drawn. One examiner's report on a Higher paper describes a question like that. "Very few candidates attempted to use trigonometry or Pythagoras' to find the length of PB. This part was often omitted." Left blank, rather than answered wrongly. Every tool this needs is one you already have, and only the shape is new.

This is video eleven of eleven in Pythagoras, Trigonometry and Coordinate Geometry. The angle in the second example is found the same way as in Finding the angle between a line and a plane, which is worth a look first if you need it.

Unfamiliar solids like these come up on Higher papers, so check how your own specification sets them.

The move in any unfamiliar solid, whatever its shape, is the same. Find a square corner, lift out the flat right-angled triangle that holds it, and solve it with a tool you know. Right, label the wedge. The near end is triangle P Q R: P is the sharp corner on the floor, Q is the square corner, and R is the top of the upright. The far end copies it as S, T and U. The line wanted runs from P, at the near end, to U, the top corner at the far end. Now, look at R U, the twelve-centimetre edge along the top, from the near end to the far end. What angle does R U make with P R, the slope of the near end? Ninety degrees. An edge that runs straight back from a face meets every line on that face at a square corner. Every angle faces its own side - look straight across. In an unfamiliar solid, find the square corner first, where an edge runs straight back from a face. Stand in it and look across, and the side you see is the one you're chasing.

Standing at R, you look across at P U. So that makes P R U a right-angled triangle, with P U as its longest side. P R isn't given, so the near-end triangle comes first. P R is its longest side, so add the squares: four squared is sixteen, and three squared is nine. Sixteen plus nine is twenty-five, and root twenty-five is five. P R is exactly five centimetres. Now triangle P R U, with five centimetres and twelve centimetres meeting at the square corner. Now finish it: how long is P U, from the five and the twelve? Thirteen centimetres. Five squared is twenty-five, twelve squared is a hundred and forty-four, and together they make a hundred and sixty-nine, whose root is thirteen. To one decimal place, write it as thirteen point nought centimetres. The nought shows the accuracy asked for.

A composite solid comes next: a square-based pyramid glued on top of a cube of side eight centimetres. The apex, the pyramid's top point, is five centimetres straight above the centre of the shared square face. The angle wanted is between one of the pyramid's sloping edges and the shared face. The cube underneath, though, changes nothing. The shared face is the plane, and the pyramid sitting on it is all that matters. Which line on the shared face lies directly under the sloping edge? The line from the edge's corner to the centre of the face, because the apex sits straight above that centre. That line is half the diagonal of the eight-centimetre square, and the drop from the apex is the five-centimetre height. The triangle is that half-diagonal, the height and the sloping edge, with theta at the corner. Half the diagonal comes first. From the centre, a corner is four centimetres across and four along, so four squared plus four squared is sixteen plus sixteen, thirty-two. Half the diagonal is root thirty-two centimetres. It isn't a whole number, so write it as root thirty-two and don't round it. The height is opposite theta, and half the diagonal is adjacent, so the ratio is tan. Tan theta equals five over root thirty-two. Press shift, tan, five, divide, root thirty-two, close the bracket, equals. The display reads forty-one point four seven two nine three four three one. To one decimal place, the edge makes an angle of forty-one point five degrees with the shared face. A student rounds early: root thirty-two becomes five point seven, and tan theta becomes five over five point seven. That comes out at forty-one point two six, which rounds to forty-one point three. Why does that student's answer miss forty-one point five? Rounding up made the bottom of the fraction too big. A bigger bottom means a smaller tan, so the angle shrinks just enough to round down. An examiner's report on a three-D trigonometry question records this. "Premature rounding of this answer before using sine inverse, cos inverse or attempting the cosine rule led to an inaccurate final answer and a loss of a mark." The fix is the habit this video has kept all the way through: write each in-between length as a root, or keep the full display, and round only the last answer.

The door stop gave up its thirteen centimetres once P R U was lifted out. Now, some unfamiliar shapes of your own. In an unfamiliar solid, what do you look for first? A square corner, where an edge runs straight back from a face, and the flat triangle that holds it. Try a prism: ends with legs six and eight centimetres, length twenty-four. Longest diagonal? Twenty-six centimetres. The end's sloping side is ten, and ten squared plus twenty-four squared is six hundred and seventy-six, whose root is twenty-six. Now a pyramid six high on a cube of side six. What angle does an edge make with the shared face? About fifty-four point seven degrees. Centre to corner is root eighteen, so tan theta is six over root eighteen.

If you've nailed it, a thumbs-up closes off the chapter's final video for good. If not, bookmark it and come back after a few days on something else. The door stop looks friendlier the second time you meet it.

That completes Pythagoras, Trigonometry and Coordinate Geometry. Next chapter: Collecting and Representing Data, starting with Sampling methods, bias and questionnaire design.

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