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
- 19 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 19
Visual introductionPicture this
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.
In a nutshell
Reason geometrically and calculate lengths, angles, areas and volumes.
Learning cycle
Part 2 of 19
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 19
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.
Reset break
Part 4 of 19
Pause
That's sitting 1 of 5 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.
Explore the idea
Part 5 of 19
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 6 of 19
Quick check
Part 7 of 19
Reset break
Part 8 of 19
Pause
That's sitting 2 of 5 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 9 of 19
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 10 of 19
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 11 of 19
Reset break
Part 12 of 19
Pause
That's sitting 3 of 5 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.
Quick check
Part 13 of 19
Explore the idea
Part 14 of 19
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 15 of 19
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
Reset break
Part 16 of 19
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 19
Game · Recall cards
The hypotenuse is
Card 1 of 4
Mastery quiz
Part 18 of 19
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 19 of 19
Lesson round-up
Ready when you are
Quiz score
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Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaPart 1 of 19 · Watch & discover