Core 2 Flashcards
Remainder Theorem
When a polynomial f(x) is divided by (ax - b) the remainder is given by f(b/a)
Factor Theorem
If (x-a) is a factor of f(x) then:
> f(a) = 0
> x=a is a root/solution to the equation f(x) = 0
Sine Rule
a/sinA = b/sinB = c/sinC
OR
sinA/a = sinB/b = sinC/c
Cosine Rule
a² = b² + c² - 2bc cosA
OR
cosA = (b²+c²-a²) / (2bc)
Area of a Triangle
1/2ab*sinC
The Ambiguous Case
When the angle you are finding is bigger than the angle you are given there are two possible results as two triangles can be drawn with the given information ( two sides, one angle)
In general:
sin(180-x) = sinx
Logs
x = loga n x equals log base a of n Means a^x = n a to the power x equals n
Logs
Multiplication Law
loga x + loga y = loga xy
Log base a of x plus log base a of y equals log base a of xy
Logs
The Division Law
loga x - loga y = loga (x/y)
log a of x minus log a of y equals log a of x/y
Logs
Changing Bases
loga x = (logb x) / (logb a)
Log base a of x equals log b base of x divided by log base b of a
Logs
Changing Bases - Special Case
loga b
loga b = (logb b) / (logb a) = 1 / (logb b)
log base a of b equals 1 divisions sed by log base b of b
Midpoint of a Line
Line (x1,y1) -> (x2,y2)
Midpoint = (x1+x2 / 2, y1+y2 / 2)
Distance Between Two Points On A Line
√[(x1-x2)² + (y1-y2)²]
Equation of a Circle
(x-a)² + (y-b)² = r²
Equation of a Circle
Centre
(a,b)
Equation of a Circle
Radius
r
360° In Radians
2π radians
180° In Radians
π radians
Converting from Radians to Degrees
multiply by 180/π
Converting from Degrees to Radians
Divide by 180/π
Length of an Arc When Angle is in Radians
l = rθ
l = length of arc r = radius of circle θ = angle in radians
Geometric Series
General Term
a r^n-1
a= first term r = common ratio n = term number
Sum of a Geometric Series
Sn = a(1-r^n) / r-1
Sn = sum a = first term r = common ratio n = number of terms
Geometric Series
Definition
To get from one number to the next, multiply by the same number each time, the common ratio
Sum to Infinity of a Geometric Sequence
If -1< r >1 the series is convergent and the sum to infinity exists
This means that r^n tends to 0 as n tends towards infinity
In this case: SumToInfinity = a / 1-r
Otherwise the series is divergent and the sum to infinity does not exist
Circle Triangle
sinθ
sinθ = y/r
Circle Triangle
cosθ
cosθ = x/r
Circle Triangle
tanθ
tanθ = y/x
Types of Stationary Point
Maximum Point
Minimum Point
Point of Inflection
d²y/dx²
Minimum Point
d²y/dx² > 0
d²y/dx²
Maximum Point
d²y/dx² < 0
d²y/dx²
Point of Inflection
d²y/dx² = 0
Stationary Points
Gradient
f’(x) = 0
Trigonometrical Identities
tanθ = sinθ / cosθ
sin²θ + cos²θ = 1
sin 30
1 / 2
cos 30
√3 / 2
tan 30
√3 / 3
sin 60
√3 / 2
cos 60
1 / 2
tan 60
√3
sin 45
√2 / 2
cos 45
√2 / 2
tan 45
1
Trapezium Rule
Area Under Line
≈ 1/2 h [ y0 + 2( y1 + y2 + … + yn-1) + yn]
