# Core 2 Key Points

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- Created by: LShan261
- Created on: 25-04-15 18:50

## Algebra and functions

- If f(x) is a polynomial and f(a) = 0, then (x-a) is a factor of f(x)
- If f(x) is a polynomial and f(b/a) = 0, then (ax-b) is a factor of f(x)
- If a polynomial f(x) is / by (ax-b) then the remainder is f(b/a)

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## The sine and cosine rule

- Sine rule
- a/sinA = b/sinB = c/sinC
- 2 angles & length of opposite side --> unknown side
- 2 sides & opposite angle --> unknown angle

- Cosine rule
- a^2 = b^2 + c^2 - 2bccosA
*or*b^2 = a^2 + c^2 - 2accosB*or*c^2 = a^2 + b^2 - 2abcosC - 2 sides & angle between --> unknown side
- 3 lengths --> unknown angle

- a^2 = b^2 + c^2 - 2bccosA
- Rearranged form of cosine rule
- unknown angle
- cosA = b^2 + c^2 - a^2/2bc
*or*cosB = a^2 + c^2 - b^2/2ac*or*cosC = a^2 + b^2 - c^2/2ab

- area of a triangle
- 1/2absinC
- length of 2 sides (a and b) & angle C

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## Exponential and logarithms

- function y = a^x, or f(x) = a^x --> a = constant --> exponential function
- logan = x --> a^x = n --> a = base
- loga1 = 0
- logaa = 1
- log10x --> logx
- Laws of logarithms
- logaxy = logax + logay --> multiplication law
- loga(x/y) = logax - logay --> division law
- loga(x)^k = klogax --> the power law

- From the power law
- loga(1/x) = -logax

- Solve equation e.g. a^x = b by 1st taking logarithms (to base 10) of each side
- Change of base rule
- logax = logbx/logba

- From the change of base rule
- logab = 1/logba

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## Coordinate geometry in the (x,y) plane

- Mid-point of (x1,y1) and (x2,y2)
- (x1 + x2/2, y1 + y2/2)

- distance d between (x1, y1) and (x2, y2)
- Equation of the circle centre
- (x-a)^2 + (y - b)^2 = r^2

- Chord = line that joins two points on circumference of a circle
- The perpindicular from the centre of circle to chord --> bisects cord
- Angle in a semicircle --> right angle
- Tangent = line that meets circle @ 1 point only
- Angle btwn tangent & radius = 90 degrees

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## The binomial expansion

- use Pascal's Triangle to multiply out bracket
- The binomial expansions is

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## Radian measure & its application

- If arc AB has length, r, then angle AOB is 1 radian (1c or 1rad)
- Radian = angle subtended @ centre of circle by an arc whose length is = to that of the radius of the circle
- 1 radian = 180 degrees/π
- Length of an arc
- l = rØ (Ø = theta)

- Area of a sector
- 1/2r^2Ø

- Area of a segment
- 1/2r^2(Ø - sinØ)

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## Geometric sequences and series

- next term --> multiply by common ratio
- nth term
- ar^n-1 --> a=1st term r=common ratio

- Sum to n terms
- Sn = a(1-r^n)/1-r
*or*Sn = a(r^n - 1)/r-1

- Sn = a(1-r^n)/1-r
- Sum to infinity
- S∞ = a/1-r

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## Graphs of trigonometric functions

- the x-y plane / into quadrants
- for all values of Ø, definitions of sin Ø, cos Ø and tan Ø are taken to be
- sinØ = y/r
- cos Ø = x/r
- tan Ø = y/x
- x and y = coordinates r = radius

- 1 st quadrant --> Ø = acute
**A**ll functions = +ve

- 2nd quadrant --> Ø = obtuse
**S**ine = +ve

- 3rd quadrant --> Ø = reflex
- 180 degrees < Ø < 270 degrees
- Tangent = +ve

- 4th quadrant --> Ø = reflex
- 270 degrees < Ø < 360 degrees
**C**osine = +ve

- Sine & cosine have a period of 360 degrees (2π rad) --> tangent period of 180 degrees (π rad)

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## Differentiation

- Increasing function f(x) in interval (a.b). f'(x) > 0 in the interval a≤x≤b
- Decreasing function f(x) in the interval (a,b), f'(x) < 0 in the interval a≤x≤b
- Maximum point = where f(x) stops increasing & begins to decrease
- Minimum point = f(x) stops decreasing & begins to increase
- Point of inflexion = where the m is @ a max/min value in the neighbourhood of points
- Stationary point = 0 m --> max/min/inflexion
- Find the coordinates of a stationary point
- find dy/dx --> solve f'(x) = 0 --> sub into y = f(x) to find corresponding values of y

- Determining the nature of a stationary point
- dy/dx = 0 & d^2y/dx^2 > 0 --> minimum point
- dy/dx = 0 & d^2y/dx^2 < 0 --> maximum point
- dy/dx = 0 & d^2y/dx^2 = 0 --> max/min/inflexion
- dy/dx = 0 & d^2y/dx^2 = 0 BUT d^3y/dx^3 ≠ 0 --> point of inflexion

- need to find the max/min point
- establish formula for y in terms of x
- differentiate
- derived function = 0
- find x then y

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## Trigonometrical identities & simple equations

- tan Ø = sin Ø/cos Ø
- sin^2Ø + cos^2Ø = 1
- sinx = k
- 1st solution --> a = sin^-1(k) (a = alpha)
- 2nd solution --> (180 degrees - a) or (π - a), if working in radians

- cosx = k
- 1st solution --> a = cos^-1(k)
- 2nd solution --> (360 degrees - a) or (2π - a)

- tanx = k
- 1st solution --> a = tan^-1(k)
- 2nd solution --> (180 degrees + a) or (π + a)

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