LearningMath
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A CAPS-aligned Grade 10–12 maths formula sheet — print it, stick it above your desk, and stop flipping through your textbook to find the sine rule.

Algebra & Equations

  • Quadratic formula
    x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Difference of squares
    a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
  • Sum/difference of cubes
    a3±b3=(a±b)(a2∓ab+b2)a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)
  • Exponent law
    am⋅an=am+n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad (a^m)^n = a^{mn}

Functions

  • Straight line
    y=mx+cy = mx + c
  • Gradient
    m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}
  • Parabola (turning point form)
    y=a(x−p)2+qy = a(x - p)^2 + q
  • Hyperbola
    y=ax−p+qy = \dfrac{a}{x - p} + q
  • Exponential
    y=a⋅bx+p+qy = a \cdot b^{x+p} + q

Trigonometry

  • Identity
    sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
  • Tangent identity
    tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}
  • Area rule
    Area=12absin⁡C\text{Area} = \tfrac{1}{2}ab\sin C
  • Sine rule
    asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}
  • Cosine rule
    a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A

Sequences & Series

  • Arithmetic nth term
    Tn=a+(n−1)dT_n = a + (n-1)d
  • Arithmetic sum
    Sn=n2[2a+(n−1)d]S_n = \dfrac{n}{2}\left[2a + (n-1)d\right]
  • Geometric nth term
    Tn=a⋅rn−1T_n = a \cdot r^{n-1}
  • Geometric sum
    Sn=a(rn−1)r−1,r≠1S_n = \dfrac{a(r^n - 1)}{r - 1}, \quad r \neq 1
  • Sum to infinity
    S∞=a1−r,−1<r<1S_\infty = \dfrac{a}{1-r}, \quad -1 < r < 1

Financial Maths

  • Simple interest
    A=P(1+in)A = P(1 + in)
  • Compound interest
    A=P(1+i)nA = P(1+i)^n
  • Future value annuity
    F=x[(1+i)n−1]iF = \dfrac{x\left[(1+i)^n - 1\right]}{i}
  • Present value annuity
    P=x[1−(1+i)−n]iP = \dfrac{x\left[1 - (1+i)^{-n}\right]}{i}

Calculus (Grade 12)

  • Derivative (first principles)
    f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}
  • Power rule
    f(x)=xn  ⟹  f′(x)=nxn−1f(x) = x^n \implies f'(x) = nx^{n-1}
  • Stationary points
    f′(x)=0f'(x) = 0

Statistics & Probability

  • Mean
    xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}
  • Standard deviation
    σ=∑(x−xˉ)2n\sigma = \sqrt{\dfrac{\sum(x-\bar{x})^2}{n}}
  • Addition rule
    P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
  • Independent events
    P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)

Analytical Geometry

  • Distance between two points
    d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
  • Midpoint
    M=(x1+x22,y1+y22)M = \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)
  • Gradient
    m=y2−y1x2−x1m = \dfrac{y_2-y_1}{x_2-x_1}
  • Parallel lines
    m1=m2m_1 = m_2
  • Perpendicular lines
    m1×m2=−1m_1 \times m_2 = -1
  • Circle (centre origin)
    x2+y2=r2x^2 + y^2 = r^2
  • Circle (centre (a,b))
    (x−a)2+(y−b)2=r2(x-a)^2 + (y-b)^2 = r^2

Euclidean Geometry

  • Angles on a straight line
    A^+B^=180∘\hat{A} + \hat{B} = 180^\circ
  • Angles at a point
    A^+B^+C^=360∘\hat{A} + \hat{B} + \hat{C} = 360^\circ
  • Angle sum of a triangle
    A^+B^+C^=180∘\hat{A} + \hat{B} + \hat{C} = 180^\circ
  • Exterior angle of a triangle
    A^ext=B^+C^\hat{A}_{ext} = \hat{B} + \hat{C}
  • Angle at centre vs circumference
    O^=2P^\hat{O} = 2\hat{P}
  • Angles in the same segment
    P^1=P^2\hat{P}_1 = \hat{P}_2
  • Cyclic quadrilateral
    A^+C^=180∘\hat{A} + \hat{C} = 180^\circ
  • Tangent-chord (tan-chord theorem)
    T^=P^inscribed\hat{T} = \hat{P}_{\text{inscribed}}

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