Year 11 Higher · Set notation and Venn diagrams
Read and write the standard notation for sets
Learn how to solve any triangle with the sine and cosine rules, apply them to bearings, recognise the sine, cosine and tangent graphs, and work precisely with standard form, set notation and loci.
Definition
The sine rule and the cosine rule connect the sides and angles of any triangle, not just right-angled ones, so one carefully drawn diagram is enough to solve a journey given as bearings. The graphs of sine, cosine and tangent show how the ratios behave for every angle: repeating waves for sin and cos, and a repeating curve with breaks for tan. Alongside them sit three skills that demand the same precision: standard form for calculating with very large and very small numbers, set notation for describing collections exactly, and loci, accurate drawings of every point obeying a distance rule.
Why Do I Have to Learn This?
Trigonometry links angles to lengths in triangles. Engineers, pilots and animators use it to work out heights and distances.
Key Rules:
- 1. Read set notation symbol by symbol: A ∪ B contains everything in A or B or both, A ∩ B contains only what is in both, and A' contains everything in the universal set that is not in A.
Remember:
- •Sine rule for an opposite pair, cosine rule for two sides and the angle between them: draw the diagram first, on bearings and loci alike.
Don’t Forget...
- •Using the cosine rule with an angle that is not between the two known sides. In a² = b² + c² - 2bc cos A, the angle A must sit between sides b and c, opposite the side a you are finding; substituting a different angle subtracts the wrong correction.
- •Taking the raw difference of two bearings as the angle inside the triangle. The interior angle at a turning point comes from the back-bearing: sailing on 040° then 100° gives an angle of 220° - 100° = 120°, not 60°.
- •Adding numbers in standard form by adding the powers, so (3 x 10⁵) + (2 x 10⁴) becomes 5 x 10⁹. Powers of 10 only combine when multiplying; to add, match the powers first: 30 x 10⁴ + 2 x 10⁴ = 32 x 10⁴, which is 3.2 x 10⁵.
Cosine
One of the three trigonometric ratios. In a right-angled triangle, cos of an angle = adjacent ÷ hypotenuse.
cos 60° = 0.5, so when the hypotenuse is 10 cm the adjacent side is 5 cm.
Worked Example:
In triangle ABC, AB = 8 cm, AC = 5 cm and the angle at A is 60°. Work out the length of BC.
You know two sides and the angle between them, so use the cosine rule: BC² = AB² + AC² - 2 x AB x AC x cos A.
Substitute the values: BC² = 8² + 5² - 2 x 8 x 5 x cos 60°.
cos 60° is the exact value 12, so BC² = 64 + 25 - 80 x 12 = 89 - 40 = 49.
Take the square root: BC = √49 = 7.
Answer: BC = 7 cm
Why It Works:
Dropping a perpendicular from one vertex splits any triangle into two right-angled ones. Applying Pythagoras to both and eliminating the shared height produces a² = b² + c² - 2bc cos A, and writing that height two ways, as b sin C and as c sin B, produces the sine rule. When the angle is 90°, cos A = 0 and the cosine rule collapses back to Pythagoras, which is why both rules agree with everything you already know.
Another Example:
A ship sails 5 km from port P on a bearing of 040°, then 8 km on a bearing of 100° to reach a buoy B. Work out the distance of the buoy from the port, giving your answer to one decimal place.
Sketch the journey with a north line at P and at the turning point Q, where the ship changes course.
Find the angle inside triangle PQB at Q: the bearing of P from Q is 040° + 180° = 220°, so the angle is 220° - 100° = 120°.
Two sides and the included angle, so use the cosine rule: PB² = 5² + 8² - 2 x 5 x 8 x cos 120°.
cos 120° = -12, so PB² = 25 + 64 + 40 = 129.
PB = √129 = 11.357..., which rounds to 11.4 km.
Answer: 11.4 km (to 1 decimal place)
Animated Example:
Free Read and write the standard notation for sets Worksheet
Download free worksheets and practise today.
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Before this: Probability (Year 10 Higher)
Let’s Practise the Concept
Step 1 of 3
Put them in order
Tap the numbers from smallest to largest.
In order so far: nothing yet