Lesson 1 · Geometry

📐 Trigonometric Ratios: SOH CAH TOA

Pythagoras links the sides of a right-angled triangle. Trigonometry links the sides to the angles, so you can find a missing side or angle from very little information.

🔤 Naming the sides

In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle. For the angle θ you are working with, the opposite side is across from θ and the adjacent side is next to it.

sin θ = opposite ÷ hypotenuse (SOH) cos θ = adjacent ÷ hypotenuse (CAH) tan θ = opposite ÷ adjacent (TOA)

Label first: decide hypotenuse, opposite and adjacent before choosing sin, cos or tan. Switching θ switches which side is opposite and which is adjacent.

🖼️ See it

Label the sides from the angle θ you are using, then pick the ratio that has the two sides you know or need.

🧮 Finding sides and angles

To find a side, rearrange the ratio. To find an angle, use the inverse function (sin⁻¹, cos⁻¹ or tan⁻¹) on your calculator, set to degrees.

Opposite = 5, hypotenuse = 10 sin θ = 5 ÷ 10 = 0.5 θ = sin⁻¹(0.5) = 30° θ = 30°, hypotenuse = 12 opposite = 12 × sin 30° = 6

Exact values to remember: sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1.

✏️ Quick check · 1 of 8

Which ratio?

You know the opposite side and the hypotenuse. Which ratio do you use?

Match the letters O and H.

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Which ratio?

You know the adjacent side and the opposite side. Which ratio do you use?

O and A.

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The hypotenuse

The hypotenuse is always:

It faces the 90° angle.

✍️ Type your answer · 4 of 8

A right-angled triangle has an angle of 30° and hypotenuse 12 cm.

What is the length of the side opposite the 30° angle, in cm?

Enter an exact answer, without rounding.

cm

Use sin = opposite ÷ hypotenuse.

✍️ Type your answer · 5 of 8

Opposite = 5 cm and hypotenuse = 10 cm.

What is the angle θ, in degrees?

Enter an exact answer, without rounding.

°

Use the inverse sine.

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A 45° angle in a right-angled triangle with adjacent side 7 m.

What is the opposite side, in m?

Enter an exact answer, without rounding.

m

tan 45° = 1.

✍️ Type your answer · 7 of 8

A ladder 10 m long leans against a wall and makes a 60° angle with the ground.

How far up the wall does it reach, to the nearest metre?

Round to the nearest metre.

m

Height is opposite the angle at the ground.

✍️ Type your answer · 8 of 8

The adjacent side is 4 and the hypotenuse is 8.

What is the angle θ, in degrees?

Enter an exact answer, without rounding.

°

Use the inverse cosine.

📝 Lesson quiz · 5 questions

Question 1 of 5

What does SOH stand for?

Question 2 of 5

tan θ equals:

Question 3 of 5

If sin θ = 0.5 then θ is:

Question 4 of 5

For the angle θ, the opposite side is:

Question 5 of 5

Your calculator must be in which mode for these questions?

Lesson 2 · Algebra

🧩 Expanding & Factorising Quadratic Expressions

Brackets multiply out into quadratic expressions, and quadratic expressions can be put back into brackets. The skill is the foundation for solving quadratic equations.

📤 Expanding double brackets

Multiply every term in the first bracket by every term in the second (FOIL: First, Outer, Inner, Last), then collect like terms.

(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 (x − 2)(x + 5) = x² + 5x − 2x − 10 = x² + 3x − 10

Watch the signs. A negative times a negative is positive: (x − 3)(x − 2) = x² − 5x + 6.

🖼️ See it

Adding the four boxes: x² + 2x + 3x + 6 = x² + 5x + 6. Factorising is this picture in reverse.

📥 Factorising x² + bx + c

To factorise, find two numbers that multiply to c and add to b. Those numbers go in the brackets. Always expand to check.

x² + 7x + 12: numbers that multiply to 12 and add to 7: 3 and 4 = (x + 3)(x + 4) x² − x − 6: multiply to −6, add to −1: −3 and 2 = (x − 3)(x + 2) x² − 9 = (x − 3)(x + 3) (difference of two squares)

✏️ Quick check · 1 of 8

Expand

Expand (x + 2)(x + 5).

Multiply each pair, then add like terms.

✏️ Quick check · 2 of 8

Expand

Expand (x − 3)(x + 3).

The middle terms cancel.

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Factorise

Factorise x² + 5x + 6.

Multiply to 6, add to 5.

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Factorise

Factorise x² − 16.

16 is 4 squared.

✍️ Type your answer · 5 of 8

Expand (x + 4)(x + 5) = x² + bx + 20.

What is the value of b?

Enter an exact answer, without rounding.

Add the two middle terms.

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Expand (x − 2)(x + 6) = x² + bx − 12.

What is the value of b?

Enter an exact answer, without rounding.

Add 6x and −2x.

✍️ Type your answer · 7 of 8

Factorise x² + 8x + 15 as (x + p)(x + q) with p smaller than q.

What is the value of q?

Enter an exact answer, without rounding.

Multiply to 15, add to 8.

✍️ Type your answer · 8 of 8

Evaluate (x + 3)(x + 4) when x = 2.

What is the value?

Enter an exact answer, without rounding.

Substitute x = 2.

📝 Lesson quiz · 5 questions

Question 1 of 5

(x + 1)(x + 1) expands to:

Question 2 of 5

Factorise x² − 5x + 6.

Question 3 of 5

Which factorisation is correct for x² + x − 6?

Question 4 of 5

The best way to check a factorisation is to:

Question 5 of 5

x² − 25 factorises to:

Lesson 3 · Probability

🎲 Sample Spaces & Mutually Exclusive Events

Listing every possible outcome makes probability questions much easier, and some events simply cannot happen together.

🎲 Sample spaces

The sample space is the list of all possible outcomes. For two events, use a table (a sample-space diagram). Then P(event) = favourable outcomes ÷ total outcomes, as long as the outcomes are equally likely.

two coins: HH, HT, TH, TT → 4 outcomes coin and dice: 2 × 6 = 12 outcomes two dice: 6 × 6 = 36 outcomes total of 7: (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) → P = 6/36 = 1/6

🖼️ See it

There are 36 equally likely outcomes. Six of them total 7, so P(total of 7) = 6/36 = 1/6.

🚫 Mutually exclusive events

Events are mutually exclusive if they cannot happen at the same time, such as rolling a 2 and rolling a 5 on one dice. Then P(A or B) = P(A) + P(B). An event either happens or it does not, so P(not A) = 1 − P(A). The expected number of times = probability × number of trials.

P(red) = 0.25, P(blue) = 0.4 → P(red or blue) = 0.65 P(green) = 1 − 0.65 = 0.35 P(seed grows) = 0.8, 50 seeds → expect 0.8 × 50 = 40

Only add when the events cannot happen together. “Even” and “a 2” overlap, so you cannot just add them.

✏️ Quick check · 1 of 7

Exclusive

Which pair of events is mutually exclusive when you roll one dice?

Can both happen in the same roll?

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Not

P(rain) = 0.3. What is P(no rain)?

The two add up to 1.

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Space

Two coins are tossed. Which list is the full sample space?

Think of a first and a second coin.

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Add

For mutually exclusive events A and B, P(A or B) equals:

Or means add here.

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A spinner lands on red with probability 0.25 and blue with 0.4. The only other colour is green.

What is P(green)?

Enter an exact answer, without rounding.

All the probabilities add to 1.

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Two dice are rolled and the scores are added.

How many of the 36 outcomes give a total of 7?

Enter an exact answer, without rounding.

List the pairs.

✍️ Type your answer · 7 of 7

P(a seed grows) = 0.8. You plant 50 seeds.

How many do you expect to grow?

Enter an exact answer, without rounding.

Probability × number of trials.

📝 Lesson quiz · 5 questions

Question 1 of 5

A bag has 3 red, 5 blue and 2 green beads. P(red or green) is:

Question 2 of 5

A coin is tossed and a dice is rolled. How many outcomes are in the sample space?

Question 3 of 5

P(win) = 0.15. What is P(not win)?

Question 4 of 5

Two dice are rolled. What is P(a double)?

Question 5 of 5

Events that cannot happen at the same time are called:

Final Challenge

🏆 10-Question Challenge Quiz

Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.

Q1 · Trigonometric Ratios: SOH CAH TOA

A right-angled triangle has hypotenuse 20 and an angle of 30°. The opposite side is:

Q2 · Trigonometric Ratios: SOH CAH TOA

In a right-angled triangle, the side opposite the right angle is the:

Q3 · Trigonometric Ratios: SOH CAH TOA

sin 30° equals:

Q4 · Trigonometric Ratios: SOH CAH TOA

A 10 m ladder leans on a wall. The angle θ with the ground has cos θ = 0.6. How far is its foot from the wall?

Q5 · Expanding & Factorising Quadratic Expressions

Expand (2x + 1)(x + 3).

Q6 · Expanding & Factorising Quadratic Expressions

Expand (x + 3)(x + 5):

Q7 · Expanding & Factorising Quadratic Expressions

Factorise x² − 9:

Q8 · Expanding & Factorising Quadratic Expressions

Factorise x² + 7x + 12:

Q9 · Sample Spaces & Mutually Exclusive Events

The probabilities of all the outcomes of an experiment add up to:

Q10 · Sample Spaces & Mutually Exclusive Events

Events that cannot happen at the same time are called: