Trigonometry: from the circle to the wave
Put a point on a circle of radius 1 and turn it by an angle θ. Its height is sin θ and its sideways position is cos θ. As θ grows, the height goes up to 1, down to −1 and back: a wave that repeats every 360° (2π radians). That is why sine and cosine describe sound, light and the swing of a pendulum.
Angles can be in radians: π radians = 180°. Always true on the circle: sin²θ + cos²θ = 1, because the point is on a circle of radius 1.
Calculus: derivative (slope) and integral (area)
The derivative tells how fast something changes. On a graph it is the slope of the tangent, the line that just touches the curve at one point. Take two nearby points, make them come closer and closer, and the slope of the line joining them settles at the slope of the tangent.
Rule to start with (power rule): if y = xⁿ then dy/dx = n xⁿ⁻¹. So for y = x² the slope is 2x, and for y = x²/4 it is x/2.
The integral adds up many thin slices to find an area under a curve. With bars: area ≈ sum of (height × width). More and thinner bars give a better answer. For y = xⁿ, ∫ xⁿ dx = xⁿ⁺¹/(n+1). The area under y = x²/4 from 0 to 4 is 4³/12 = 5.33.
Derivative and integral undo each other. This link is the fundamental theorem of calculus.
Geometry: shapes and solids
In coordinate geometry we describe shapes with numbers. The distance between (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²), and a circle with centre (0, 0) and radius r is x² + y² = r².
For solids, take a cylinder of radius r and height h = 2r. Then a cone of the same size holds one third of it and a sphere that just fits inside holds two thirds of it:
V(cylinder) = πr²h, V(cone) = πr²h/3, V(sphere) = 4πr³/3.
Archimedes found this ratio 1 : 2 : 3 more than 2,000 years ago. In the 3D you can pour the cone and the sphere into the cylinder: they fill it exactly.
Probability and statistics
Probability = favourable outcomes ÷ all equally likely outcomes. For two dice there are 6 × 6 = 36 outcomes. A sum of 7 can happen in 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so P(7) = 6/36 = 1/6, more than P(2) = 1/36. That is why the middle bar is tallest.
Statistics describes data. The mean is the average. The standard deviation tells how far values usually spread from the mean. When many small chances add up (like two or more dice), the bar chart of results forms a smooth bell curve, the normal distribution. About 68% of values lie within one standard deviation of the mean.
Try it and what to learn next
Try it: roll the dice 100 times in the 3D and predict which bar will be tallest before you look. Then roll 100 more. The more you roll, the closer the blue bars match the orange outline.
Next, go deeper with: Unit circle, Derivatives, Integration, Solid geometry and Normal distribution.
Key formulas and definitions
- sin²θ + cos²θ = 1; π rad = 180°
- d/dx (xⁿ) = n xⁿ⁻¹
- ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C
- V(cylinder) = πr²h; V(cone) = ⅓ πr²h; V(sphere) = 4⁄3 πr³
- P(event) = favourable ÷ total; mean = sum ÷ count
Worked examples
1. If θ is acute and cos θ = 0.6, find sin θ.
sin²θ + cos²θ = 1, so sin²θ = 1 − 0.36 = 0.64. sin θ = 0.8 (positive, since θ is acute).
2. Find the slope of y = 3x² + 5x at x = 2.
dy/dx = 6x + 5. At x = 2: 6 × 2 + 5 = 17.
3. Find the area under y = x² from x = 0 to x = 3.
Area = ∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9.
4. Find the volume of a sphere of radius 3 cm.
V = 4⁄3 π r³ = 4⁄3 × π × 27 = 36π ≈ 113.1 cm³.
5. Two fair dice are rolled. Find the probability that the sum is 8.
Ways to get 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 ways out of 36. P = 5/36 ≈ 0.139.
6. Find the mean and standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.
Mean = 40 ÷ 8 = 5. Differences squared: 9, 1, 1, 1, 0, 0, 4, 16 = 32. Variance = 32 ÷ 8 = 4. Standard deviation = √4 = 2.
Common mistakes
- Mixing degrees and radians on a calculator. Check the mode first.
- Forgetting to lower the power by one when you differentiate: the derivative of x³ is 3x², not 3x³.
- Forgetting the + C when you find an indefinite integral.
- Counting dice outcomes as 11 equal ones (sums 2 to 12). The sums are not equally likely: there are 36 equal outcomes.