Equations, Indices, Logarithms & Trigonometry
1. Simultaneous Linear Equations
Solving TWO equations in TWO unknowns.
Methods
| Method | How |
|---|---|
| Elimination | Make coefficients of one variable EQUAL. Add or subtract equations to ELIMINATE it. |
| Substitution | Express one variable in terms of the other from ONE equation. Substitute into the OTHER. |
| Cross-Multiplication | Use formula when equations are: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 |
Word Problems
Translate the problem into TWO equations. Solve. Check for reasonableness.
2. Indices (Exponents)
Laws (Extended)
| Law | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ |
| Power of power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ |
| Fractional exponent | a^(1/n) = ⁿ√a | 8^(1/3) = ∛8 = 2 |
| a^(m/n) | ⁿ√(aᵐ) = (ⁿ√a)ᵐ | 8^(2/3) = (∛8)² = 4 |
3. Logarithms
Definition
If aˣ = N, then logₐN = x. 'The logarithm is the EXPONENT.'
Common Log (base 10): log₁₀N = log N. Natural Log (base e): ln N.
Laws
| Law | Formula |
|---|---|
| Product | logₐ(MN) = logₐM + logₐN |
| Quotient | logₐ(M/N) = logₐM — logₐN |
| Power | logₐ(Mᵖ) = p × logₐM |
| Change of Base | logₐN = logₐb × logᵦN |
4. Trigonometry
Trigonometric Ratios (Right Triangle)
| Ratio | Formula |
|---|---|
| sin θ | Opposite / Hypotenuse |
| cos θ | Adjacent / Hypotenuse |
| tan θ | Opposite / Adjacent = sin θ / cos θ |
| cosec θ | 1/sin θ |
| sec θ | 1/cos θ |
| cot θ | 1/tan θ = cos θ / sin θ |
Standard Angle Values — MEMORISE
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
Fundamental Identity
sin²θ + cos²θ = 1 → 1 + tan²θ = sec²θ → 1 + cot²θ = cosec²θ
Solving Right Triangles
Given two sides (or one side and one acute angle), find the remaining sides/angles using trig ratios.
Simple 2D Problems
- Angle of ELEVATION: Looking UP from horizontal
- Angle of DEPRESSION: Looking DOWN from horizontal
Complementary Angles
sin(90° — θ) = cos θ. cos(90° — θ) = sin θ. tan(90° — θ) = cot θ.
