How to teach Pythagoras' theorem and trigonometry
Year 8 to Year 10 (ages 13 to 16)
Pythagoras' theorem connects the three sides of a right-angled triangle, so knowing any two gives you the third. Trigonometric ratios (sine, cosine, tangent) connect an angle to a ratio of sides, so knowing one side and one angle gives you the rest.
How to teach it
- Teach a^2 + b^2 = c^2 for finding the hypotenuse first, using whole-number triples (3-4-5, 5-12-13) so the arithmetic stays clean.
- Show the rearranged version for a missing leg: a leg squared is the hypotenuse squared minus the other leg squared.
- Introduce SOH-CAH-TOA once Pythagoras is secure, always labelling the hypotenuse, opposite and adjacent sides relative to the angle first.
- Demonstrate that scaling a triangle up (e.g. 3-4-5 to 6-8-10) keeps every trig ratio identical, since the triangles are similar.
- Apply both tools to real measuring problems (ladders, height, distance) so students choose the right tool for what is known.
Worked example
A right triangle has legs 3 and 4 c^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = 5 For the angle opposite the side of length 3: sin = 3/5 = 0.6, cos = 4/5 = 0.8, tan = 3/4 = 0.75
Common mistakes
- Assuming the hypotenuse can be any side chosen, rather than always the longest side, opposite the right angle.
- Adding the squares to find a missing leg, instead of subtracting when the hypotenuse is already known.
- Believing a bigger triangle has bigger trig ratios, confusing longer sides with a changed ratio.
- Mixing up which side is opposite versus adjacent when the reference angle changes.
Frequently asked questions
What is Pythagoras' theorem?
In any right-angled triangle, a^2 + b^2 = c^2, where c is always the hypotenuse, the longest side, opposite the right angle. Knowing any two sides lets you find the third.
What are sine, cosine and tangent?
They are three ratios of sides relative to an angle in a right-angled triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent, remembered with SOH-CAH-TOA.
Why do trig ratios stay the same for a bigger similar triangle?
Scaling a triangle up multiplies every side by the same amount, so the RATIO between any two sides is unchanged. A 3-4-5 triangle and a 6-8-10 triangle share exactly the same angles and the same sin, cos and tan values.
What year is Pythagoras' theorem and trigonometry taught?
In the Australian Curriculum, Pythagoras' theorem starts at Year 8 (AC9M8M06), with trigonometric ratios and applying both to spatial problems introduced at Year 9 (AC9M9M03, AC9M9SP01), extending through Year 10.
Practise with free worksheets
Printable worksheets with answer keys that are never wrong.