GCSE Higher Tier Math Formulas

Introduction

• The video is focused on formulas needed for the higher tier GCSE math paper.
• Previous video covers formulas needed for both foundation and higher tier; recommended to watch it as well.
• This video focuses solely on higher tier-specific formulas.

Volume of a Pyramid

• Formula: ( \frac{1}{3} \times \text{Area of the base} \times \text{Height} )
  • Example:
    • Base: 6 x 6, Height: 8
    • Calculation: ( \frac{1}{3} \times 36 \times 8 = 96 \text{ cm}^3 )

Quadratic Equation

• Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
  • Example:
    • Equation: ( 3x^2 + 5x - 4 = 0 )
    • Solutions: ( x_1 = 0.59, x_2 = -2.26 )

Sine Rule

• Formula: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )
  • Used for non-right angled triangles.
  • Example:
    • Opposite pairs: ( A = 30^\circ, a = x; B = 62^\circ, b = 8 )
    • Calculation: ( x = \frac{8 \times \sin 30}{\sin 62} = 2.8 \text{ cm} )

Cosine Rule

• Formula: ( a^2 = b^2 + c^2 - 2bc \cos A )
  • Used for non-right angled triangles when opposite pair not available.
  • Example:
    • Sides: 6, 8; Angle: 60°
    • Calculation: ( x = \sqrt{52} = 7.2 \text{ cm} )

Area of a Triangle (Using Sine)

• Formula: ( \frac{1}{2} ab \sin C )
  • Used when height of triangle is unknown.
  • Example:
    • Sides: 7, 8; Angle: 30°
    • Calculation: ( 14 \text{ cm}^2 )

Area of a Sector

• Formula: ( \frac{\theta}{360} \times \pi r^2 )
  • Example:
    • Angle: 62°, Radius: 8
    • Calculation: ( 34.6 \text{ cm}^2 )

Direct and Inverse Proportion

• Direct Proportion: ( x = ky^n )
  • Example: ( x \propto y^2 ), ( x = 18, y = 6 ), calculate x for ( y = 8 )
• Inverse Proportion: ( x = \frac{k}{y^n} )

Histograms and Frequency Density

• Formula: ( \text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} )
  • Example:
    • Frequency: 8, Class Width: 10
    • Calculation: ( 0.8 )

Equation of a Straight Line

• Formula: ( y = mx + c )
  • Gradient: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
  • Perpendicular Gradient: Negative reciprocal of original.
  • Example:
    • Points: A(1, 5), B(4, 11)
    • Perpendicular line through (6, 7): ( y = -\frac{1}{2}x + 10 )

Conclusion

• These are key formulas for the higher tier paper.
• Further detailed videos are available in the description for each topic.