Pythagoras and trigonometry
In a right-angled triangle, Pythagoras handles sides alone and trigonometry handles anything involving an angle. Choosing between them is the first decision.
Lesson overview
What you'll learn in this lesson
Reason geometrically and calculate lengths, angles, areas and volumes.
Key learning points
- • Pythagoras' theorem
- • The trigonometric ratios
- • Angles of elevation and 3D problems
This lesson at a glance
- 30 minutes
- 15 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 15
Watch first
Pythagoras and trigonometry
An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.
In a nutshell
Reason geometrically and calculate lengths, angles, areas and volumes.
Learning cycle
Part 2 of 15
Learning cycle 1 of 2
Part 1 · Pythagoras' theorem
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 15
Learn
Pythagoras' theorem
a² + b² = c², where c is the hypotenuse, the longest side, opposite the right angle. To find a shorter side, subtract rather than add. The theorem also proves whether a triangle is right-angled: if the squares balance, it is.
Explore the idea
Part 4 of 15
Learn
The trigonometric ratios
Label the sides relative to the given angle: opposite, adjacent and hypotenuse. Then sin θ = opp ÷ hyp, cos θ = adj ÷ hyp and tan θ = opp ÷ adj. To find an angle, use the inverse function on your calculator. Label the triangle before choosing a ratio and errors largely disappear.
Quick check
Part 5 of 15
Quick check
The hypotenuse is
Quick check
Part 6 of 15
Quick check
To find a shorter side using Pythagoras you
Reset break
Part 7 of 15
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 8 of 15
Learning cycle 2 of 2
Part 2 · Angles of elevation and 3D problems
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 9 of 15
Learn
Angles of elevation and 3D problems
The angle of elevation is measured from the horizontal upwards, and depression downwards. In three dimensions, identify a right-angled triangle inside the solid, often through a diagonal of the base, and solve it as a two-dimensional problem. Sketching that triangle separately is worth the thirty seconds.
Quick check
Part 10 of 15
Quick check
If you know the opposite and adjacent sides, use
Quick check
Part 11 of 15
Quick check
The angle of depression is measured from
Explore the idea
Part 12 of 15
Worked example
Worked answer: a ladder 5 m long leans against a wall with its base 1.4 m out. Find the angle with the ground (3 marks)
The ladder is the hypotenuse and the 1.4 m is adjacent to the required angle, so use cosine (1). cos θ = 1.4 ÷ 5 = 0.28 (1). θ = cos⁻¹(0.28) = 73.7° to 1 decimal place (1). Choosing sine here is the classic error: 1.4 m is along the ground, adjacent to the angle, not opposite it.
Challenge round
Part 13 of 15
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Mastery quiz
Part 14 of 15
Marked quiz
End of lesson quiz: Pythagoras and trigonometry
4 questions, marked with the reasoning shown. No timer.
1. The hypotenuse is
2. To find a shorter side using Pythagoras you
3. If you know the opposite and adjacent sides, use
4. The angle of depression is measured from
Lesson round-up
Part 15 of 15
Lesson round-up
Ready when you are
Quiz score
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Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 15 · Watch & discover