Pythagoras and Trigonometry
Pythagoras and the right-angled triangle
Year 9 introduced two tools for the right-angled triangle, and this unit puts them to work on practical problems: heights, distances, directions. The first is the theorem of Pythagoras. It says that in any right-angled triangle, the square built on the longest side, the hypotenuse opposite the right angle, has an area equal to the two squares built on the other sides added together. Written with letters, a² + b² = c², where c is the hypotenuse. With legs of 3 and 4, the squares are 9 and 16, and their sum, 25, is the square on the hypotenuse, which is therefore 5. This relationship lets you find the third side of a right-angled triangle whenever the other two are known, a tool used constantly in building, navigation and design.
Calculating a side with Pythagoras
Using the theorem is direct. To find the hypotenuse, square each of the two shorter sides, add the results, and take the square root. For legs of 6 and 8, the hypotenuse squared is 36 + 64 = 100, so the hypotenuse is √100, exactly 10. The method also works in reverse to find a shorter side: if the hypotenuse and one leg are known, you subtract the square of that leg from the square of the hypotenuse before taking the root. Because the formula involves squares and a square root, an answer is often an exact surd rather than a whole number, and knowing when to leave it exact and when to round is part of using appropriate units and accuracy.
Trigonometry: linking sides to angles
Pythagoras connects the three sides, but it says nothing about the angles. Trigonometry fills that gap through three ratios of the sides, each tied to one of the acute angles. Labelling the sides relative to an angle as opposite, adjacent, and hypotenuse, sine = opposite/hypotenuse, cosine = adjacent/hypotenuse and tangent = opposite/adjacent. The memory phrase SOH CAH TOA captures all three. The remarkable fact that makes trigonometry work is that these ratios depend only on the angle, not on the size of the triangle: every right-angled triangle with a given angle has the same sine, cosine and tangent for it, so the ratios can be tabulated and used for any such triangle.
Finding an unknown side
The trigonometric ratios let you find a missing side when an angle and one side are known. The trick is to choose the ratio that connects what you know to what you want. Suppose a right-angled triangle has a 30° angle and a hypotenuse of 10, and you need the side opposite the angle. Sine links opposite and hypotenuse, so sin 30° = opposite/10, which rearranges to opposite = 10 × sin 30°. Since sin 30° is exactly 0.5, the opposite side is 5. Choosing sine, cosine or tangent according to which sides are involved, then rearranging, solves a vast range of practical problems, from the height of a tree to the slope of a ramp.
Finding an unknown angle
Trigonometry also runs the other way: when two sides are known, it can recover the angle between or beside them. Here the inverse ratios are needed, written sin−1, cos−1 and tan−1. If the side opposite an angle is 3 and the adjacent side is 4, then tan θ = 3/4 = 0.75, and the angle itself is tan−1(0.75), about 36.9°. The inverse ratio simply asks the reverse question: not what is the ratio for this angle, but what angle gives this ratio. Together, Pythagoras for sides and trigonometry for the link between sides and angles make the right-angled triangle a complete and powerful measuring instrument.
Angles of elevation and depression
Most real uses of trigonometry start from a horizontal line of sight. The angle of elevation is the angle you look up from the horizontal to see something higher, such as the top of a tree or a tower; the angle of depression is the angle you look down from the horizontal to something lower, such as a boat seen from a clifftop. Both are measured from the horizontal, and because the horizontal at your eye and the horizontal at the object are parallel, an angle of elevation up to an object equals the angle of depression back down from it.
These problems become right-angled triangles. To find the height of a tree when you stand 20 metres away and the angle of elevation to its top is 35°, the height is the opposite side and the 20 metres is the adjacent side, so tan 35° = height/20, giving a height of 20 × tan 35°, about 14 metres. The same reasoning with an angle of depression finds how far away a boat is from the foot of a cliff of known height: from a 40 m clifftop with an angle of depression of 25°, the boat is 40 ÷ tan 25°, about 85.8 m, out. Choosing the ratio that links the side you know to the side you want is the whole skill.
Direction: bearings
Direction problems use bearings. A true bearing is an angle measured clockwise from north and written with three figures: north is 000°, east 090°, south 180° and west 270°, so a bearing of 050° points between north and east, 50° round from north. A journey on a bearing becomes a right-angled triangle once you draw the north-south line through the start: the path is the hypotenuse, and the distances north or south and east or west are the two legs. Sailing 10 km on a bearing of 050°, a boat ends 10 × cos 50° ≈ 6.43 km north and 10 × sin 50° ≈ 7.66 km east of where it started, and Pythagoras checks the legs: 6.43² + 7.66² ≈ 100 = 10². For a bearing past 090°, first find the acute angle the path makes with the north-south line: 120° is 60° from south towards east.