Pythagoras and Trigonometry in Right-Angled Triangles
ACARA v9 CONTENT DESCRIPTION “solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles”
Builds on: Real Numbers: Rational and Irrational (AC9M9N01). This unit builds on the real number system, square roots and surds, and on ratio. Pythagoras' theorem and trigonometry are foundational for the spatial reasoning in the Space strand and for any later work in geometry and physics.
The most useful triangle
The right-angled triangle is one of the most useful shapes in all of mathematics, and this unit gathers several powerful tools for working with it: the angle facts that constrain it, the ideas of scale and similarity that relate triangles of the same shape, Pythagoras' theorem for its sides, and trigonometry for linking its sides to its angles. Together these let you find an unknown length or angle in a huge range of practical problems, from measuring a building's height to checking that a corner is truly square.
Naming the sides
Every trig ratio rests on three names: the hypotenuse is the longest side opposite the right angle, while the opposite and adjacent sides are named relative to the acute angle you choose.
Choosing the lower acute angle: the vertical leg faces it, so it is the opposite, and the horizontal leg touches it, so it is the adjacent. The hypotenuse stays the longest side opposite the right angle.
Angle facts and complementary angles
Start with the angle facts, because they are the simplest constraint. The three angles of any triangle add to 180°. In a right-angled triangle one angle is already 90°, so the remaining two must add to 90°; they are said to be complementary. This means that knowing one of the acute angles immediately gives the other, a small fact that saves a great deal of work later.
Angles add to 180, the acute pair to 90
Knowing one acute angle gives the other for free: the right angle uses ninety degrees, leaving exactly ninety to share between the two acute angles.
Set the lower acute angle to 37° and the top angle is forced to 53°, because the right angle already uses 90° and the three must total 180°. The two acute angles always add to 90°, so each one fixes the other.
Scale and similar triangles
Scale and similarity come next. Two triangles are similar when they have the same shape but possibly different sizes, which happens exactly when their angles match. Corresponding sides of similar triangles are then in a fixed ratio called the scale factor. A three, four, five triangle enlarged by a scale factor of two becomes a six, eight, ten triangle, and every pair of corresponding sides shares the ratio two, while the angles stay exactly the same. This is the engine behind indirect measurement: if a two metre stick casts a three metre shadow while a tree casts an eighteen metre shadow, the triangles are similar, the scale factor is six, and the tree must be twelve metres tall.
Similar triangles and scale factor
Similar triangles have identical angles and sides in a fixed ratio; doubling a 3-4-5 triangle gives a 6-8-10 triangle, scale factor two, with the angles unchanged.
At scale factor 2 the 3, 4, 5 triangle enlarges to 6, 8, 10. The ratios 6:3, 8:4 and 10:5 all equal 2, so every side multiplies by the same factor while the angles never change.
Pythagoras' theorem for the sides
Pythagoras' theorem connects the three sides of a right-angled triangle through their squares. It states that the square of the hypotenuse, the longest side opposite the right angle, equals the sum of the squares of the other two sides; with hypotenuse c and shorter sides a and b, c² = a² + b². So if the two shorter sides are 3 and 4, the hypotenuse squared is 9 + 16 = 25, making the hypotenuse 5. Certain whole-number combinations like 3, 4, 5 and 5, 12, 13 recur so often they are worth memorising. The theorem also runs backwards to find a shorter side: if the hypotenuse is 13 and one side is 5, the missing side squared is 169 − 25 = 144, so that side is 12.
Pythagoras: squares on the sides
The classic area picture: build a square on each side of a right-angled triangle and the two smaller areas always add up to the largest.
For the 3, 4, 5 triangle the squares on the legs have areas 9 and 16, and 9 + 16 = 25, the area of the square on the hypotenuse. That is Pythagoras: 3 squared + 4 squared = 5 squared, so the hypotenuse is 5.
When the hypotenuse is irrational
Most triangles do not have whole-number sides, and here the earlier work on real numbers matters. A right-angled triangle with both short sides equal to 1 has a hypotenuse whose square is 2, so the hypotenuse is √2, an irrational number approximately 1.41 but never exactly that decimal. Leaving such an answer as √2, or simplifying a surd like √8 to 2√2, keeps it exact; the decimal is only an approximation and should not be written with an equals sign.
An irrational hypotenuse
When the sides do not form a neat whole-number triple, the hypotenuse is often irrational; the exact answer keeps the root sign and the decimal is only an approximation.
With legs 1 and 1, the hypotenuse squared is 1 + 1 = 2, so the hypotenuse is exactly √2. That is irrational: √2 ≈ 1.41, but the decimal never lands exactly, so the exact answer keeps the root sign.
Trigonometry: SOH CAH TOA
Trigonometry is the final and most powerful tool, linking the angles of a right-angled triangle to the ratios of its sides. For a chosen acute angle, the side touching it is the adjacent, the side facing it is the opposite, and the longest is the hypotenuse. The three ratios are sine = opposite/hypotenuse, cosine = adjacent/hypotenuse and tangent = opposite/adjacent, captured by the memory aid SOH CAH TOA. A few special values are exact: sin 30° is exactly ½, and tan 45° is exactly 1. Others, like sin 45° at about 0.707, are irrational.
Trig finds a side
To find a side, pick the ratio that links the side you want to the side you know: here the opposite is the hypotenuse times the sine of the angle.
With a 30° angle and a hypotenuse of 10, the opposite side is the hypotenuse × the sine of the angle: 10 × sin 30°. Because sin 30° = 1/2 exactly, the opposite side is exactly 5. Sine works here because sine = opposite / hypotenuse links the angle to those two sides.
Finding sides and angles
These ratios solve two kinds of problem. To find a side, multiply by the ratio: in a triangle with a 30° angle and a hypotenuse of 10, the opposite side is 10 × sin 30° = 5. To find an angle, use the inverse functions: if the opposite is 3 and the adjacent is 4, the tangent of the angle is 3/4 = 0.75, so the angle is tan−1(0.75), about 37°. Choosing the right ratio depends only on which two sides you know and which one you want.
Quick self-check
1. In a right-angled triangle the two shorter sides are 6 and 8. What is the hypotenuse?
2. A right-angled triangle has hypotenuse 13 and one shorter side 5. What is the other shorter side?
3. A right-angled triangle has both shorter sides equal to 1. Its hypotenuse is:
4. Which ratio equals opposite divided by hypotenuse?
5. In a right-angled triangle a 30° angle has a hypotenuse of 10. The side opposite the 30° angle is: