Formulas

Formulas

1. Further Trigonometry

   • Right-Angled Trig
     • Sin θ = O/H θ =sin-1(O/H)
     • Cosθ = A/H θ=Cos-1 (A/H)
   • Further Trig
     • Cos Rule
       • Side: a² = b² + c² - 2bc cosA
       • Angle: cosA = (b² + c² - a²)/2bc
       • USE WHEN: 3 sides + 1 angle
     • Sin Rule:
       • Sin rule: a/SinA = b/SinB = c/SinC
   • Complimentary Ratios:
     • Sin alpha = cos(90-alpha)
     • Cos alpha = sin(90-alpha)
     • Tan alpha = (sin alpha/cos alpha)
   • Supplementary Ratios
     • Sin(180-θ) = Sinθ
     • Cos(180-θ)= - cosθ
     • Tan(180-θ)= - tanθ
   • Area of a Triangle
     • A = 1/2*a*b*sinC (2 sides + included angle)

2. Quadratic Equations

   • Quadratic Formula
     • x = (-b +/- √b² - 4ac)/2a
   • Discrminant (Δ)
     • For finding amount of possible solutions.
       • Δ > 0 = 2
       • Δ = 0 = 1
       • Δ < 0 = none
     • b² - 4ac

3. Coordinate Geometry

   • Distance
     • d = √(x₂- x₁)² + (y₂- y₁)²
   • Midpoint
     • MP = (x₁+ x₂/ 2 , y₁+ y₂/2 )
   • Gradient
     • m = (y₂- y₁) / (x₂- x₁)
   • Parallel: Same gradient
   • Perpendicular: negative reciprocal
   • y-int: Let x = 0
   • x-int = Let y = 0
   • General form of a linear equation: ax + by + c = 0
   • Point Gradient
     • WIth the gradient of the line and points it passes
     • y - y₁= m ( x - x₁)
   • Two point
     • 2 points are given
     • (y - y₁) / (x - x₁) = (y₂- y₁) / (x₂- x₁)

4. Parabolas

   • Axis of Symmetry
     • x = h (x coordinate of the vertex)
     • x = -b / 2a
     • x = x-int + x-int / 2
   • x-ints: factorise

5. Consumer Arithmetic

   • Simple Interest
     • I = PRT
       • I = Interest P = Principa lR = rate of interest as a decimal&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;T = time periods
   • Compound Interest
     • A = P(1 + r)^n
   • Depreciation
     • A = P(1 - r)^n
   • Balance owing = cost - deposit
   • Total repaid = Balance + interest
   • Total price = Total repayments + deposit
   • Credit Cards
     • Closing Balance = Opening + purchases - payments

6. Surface Area and Volume

   • Surface Area
     • Prism
       • Sum of the area of the faces. Draw a net.
     • Cylinder
       • SA = 2 π r² + 2π r h
     • Composite Shapes
       • Write in words what needs to be found
     • Rhombus/Kite
       • A = xy/2
     • Trapezium
       • A = 1/2*h*(a+b)
     • Parallelogram
       • A = bh (h = perpen.)
     • Spheres
       • SA = 4π r²
     • Hemispheres
       • SA = 2π r²
     • units²
     • Cones
       • SA = π r² + π r l (l = slant height)
   • Volume
     • units³
       • Prism
         • V = Ah
     • Cones
       • V = 1/3 Ah
     • Cylinders
       • V = π r² h
     • Spheres
       • 4/3 π r³
   • For weird cut circles: n/360 x π r²

7. Functions

   • Inverse Functions
     • 1. Swap x and y 2. Make y the subject
     • Reflect along y=x Swap x & y pts
   • Line Tests
     • Vertical line test
     • Horizontal line test
       • To check whether the inverse of a graph is a function
   • Function: Each x value only has 1 y value

8. Further Graphs

   • Effects to graphs
     • h(2x)
       • horizontal -><-
     • h(0.5x)
       • horizontal <-->
     • 2h(x)
       • vertical stretch
     • -h(x)
       • vertical flip
     • h(-x)
       • horizontal flip
   • Effects to asymptotes
     • -f(x)
       • y= 2-->y= -2
     • f(-x)
       • x= 3-->x= -3
     • f(x-2)
       • x= 3--> x=5 (opp.)
     • f(x)-2
       • y=2 --> y=0
     • f(2x)
       • x=100 --> x=50
     • 2f(x)
       • y=50 --> y=100
   • in bracket: x (horizontal)
   • Out of bracket: y (vertical)

9. Deductive Geometry

   • Congruence
     • SSS
     • SAS
     • AAS
     • RHS
   • Similarity
     • Scale Factor
       • sf = image length/original length
     • SSS
     • AA
     • RHS
     • SAS

10. Surds

    • √ab = √ a*√ b
    • (√5)² = √5*√5

11. Logarithms:

    • Log Laws
      • 1) Multiplication inside the log can be turned into addition outside the log, and vice versa.
      • 2) Division inside the log can be turned into subtraction outside the log, and vice versa.
      • 3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.
      • Log a a = 1
      • Log a 1 = 0

12. Index Laws

    • a^m x a^n = a^m+n
    • a^m / a^n = a^m-n
    • (a^m)^n = a^m*n
    • a^1/m = m√a
    • a^0 = 1