Download Trigonometry Sine Rule = = = ( ) Area of Triangle

Core 1 and Core 2
Trigonometry
Triangle ABC
Sine Rule
=
=
can be used upside down if trying to find an angle
Important that side a is opposite angle A and side b opposite angle B and side c
opposite angle C
Cosine Rule is given in formulae sheets but you need to be able to
rearrange to get as the subject
=
Area of Triangle
(
)
important that the angle C is the angle between the two sides a and b
+ = = = ᵒ
Radians
Sector Area
Arc length
Ɵ
Ɵ
Basic Trigonometry SOHCAHTOA
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!!
= = = "#!
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Page 1
Core 1 and Core 2
= (& − ) = (& − )
() = ()( + )
= ( − ) (−) = ( + )
= (*+ − )
(−) = Useful Trigonometric Triangles
9
Just remember that sin 30ᵒ = : and you can
work out the rest using Pythagoras’
Theorem and basic trigonometry
1
60ᵒ
Just remember that tan 45ᵒ = 1 and you can
work out the rest using Pythagoras’
Theorem and basic trigonometry
2
1
√
45ᵒ
30ᵒ
45ᵒ
√*
√*
*ᵒ = *ᵒ = () *ᵒ =
√*
1
34ᵒ = 34ᵒ =
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() 34ᵒ = √
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Core 1 and Core 2
Solving quadratics
factorise, complete the square or quadratic equation
Quadratic Equation
>? + @? + A = 0
:
;=
±= 3
Completing the square
we can use this to work out where the turning points are and to help sketch
the curve or to solve quadratic equations.
Eg 4? : − 16? + 5 = 4(? : − 4?) + 5 = 4((? − 2): − 2: ) − 5
4((? − 2): − 4) − 5 = 4(? − 2): − 21
Turning point (2, -21)
Line of symmetry ? = 2
Hidden Quadratics
By using substitution we can change an equation into a quadratic that can be
solved.
eg D :E + 6D E + 8 = 0 Let G = D E
so we have G : + 6G + 8 = 0
H
I
J
I
J
I
or ? + 2? − 8 = 0
let G = ? both of which can be solved
so we have G : + 2G − 8 = 0
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Core 1 and Core 2
Laws of logarithms
KL ; = # ⇔ # = ;
KL = KL = KL ; + KL # = KL ;#
;
KL ; − KL # = KL N O
#
PKL ; = KL ;P
Change of base
KL = KL Solving equations with logs
Eg G = 5E take logs of both sides
QRS9T G = QRS9T (5E ) = ? QRS9T 5
UVW Y
? = JX could also have taken logs to base 5 and then would get the same
UVWJX Z
answer ? = UVW[ Y
UVW[ Z
= QRSZ G
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Core 1 and Core 2
Calculus
Differentiation
multiply by power and reduce power by 1
# = ; $#
$;
= ;
\ ; $; = ; + Integration
increase power by 1 and then divide by new power
\ ; $; = ] ; + ;
Area under curve \; #$; between curve and ? axis
Area under curve
_(
^ ;) =
#
\# ;$# between curve and y axis
$#
$;
__ (
^ ;) = $ #
$;
Stationary Points
Minimum, maximums and points of inflection
Differentiate and set =0 and solve to find ?
To investigate whether a minimum or maximum differentiate again to get
substitute in the ? value and if positive then minimum, if negative then
maximum, if 0 then a point of inflection
Increasing function
Where the gradient is positive (sloping upwards)
Decreasing function
Where the gradient is negative (sloping downwards)
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`HE
`Y H
,
Core 1 and Core 2
Coordinate Geometry
Equation of a line
# − # = a(; − ; )
If have gradient m and a point (?9 , G9 ) that the line goes through
# #
Gradient of line
; ;
If have two points on the line (?9 , G9 ) and (?: , G: )
Equation of a circle
Centre (a, b) and radius r
(; − ) + (# − ) = ; ; # #
Midpoint
If have two points on the line (?9 , G9 ) and (?: , G: )
(
,
)
=(; − ; ) + (# − # )
Length of a line
If have two points on the line (?9 , G9 ) and (?: , G: )By Pythagoras’ Theorem
Normal is perpendicular to Tangent
The gradients are the negative reciprocal of each other
:
eg gradient tangent − then gradient normal is
c
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c
:
Page 6
Core 1 and Core 2
Sequences
Formulae given in booklet but do need to recognise if a geometric sequence or
arithmetic
Geometric
each term is a constant multiple of the previous term
Arithmetic
each term is a constant addition to the previous term
Convergent
9 9 9
eg , , ,
9
: d e 9f
each term gets closer and closer to a number
,…
Divergent
each term does not converge
eg 7, 15, 23, 31, …
Periodic
the terms start repeating
eg 3, 2, 4, 5, 3, 2, 4, 5, 3, 2, 4, 5,… has period 4
Period is number of terms before the sequence repeats
∑i ^()
sum of terms from r = 1 up to r = n
eg
− 1) = 3 + 7 + 11 + 15 + 19 = 55(sum of all the terms from r =
1 up to r = 5)
∑Zmi9(4j
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Core 1 and Core 2
Indices
na = a ÷ a = a ( )a = a = a
=
= √
a
a
= p q = r √s
Polynomials
Factor Theorem
If ^() = then ; = is a root and (; − ) is a factor
Remainder Theorem
When ^(;) is divided by (; − ) the remainder is ^()
To sketch a cubic polynomial try to factorise
(? − >)(? − @)(? − A) = 0 will cross the ? axis at a, b and c
Surds
√;
√$ = √$
√$
=
(
√$)
(
√$)(
√$)
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Core 1 and Core 2
Quadratic inequalities
solve (? − >)(? − @) < 0 Rj(? − >)(? − @) > 0
solve where this equals 0 (when ? = >Rj? = @)
and then see whether we want the bit between a and b or the bits either side
A positive quadratic has a ∪ shape so will be below the ? axis between a and b
(; − )(; − ) < < ; < (; − )(; − ) > ; < $; > Transformations
^(;) + translation of a units in the positive y direction
^(; + )translation of a units in the negative ? direction
^(;) stretch of scale factor a parallel to the y axis
^(;) stretch of scale factor w9 parallel to the ? axis
Discriminant − 3
Part of the quadratic equation. Tells us how many real roots there are.
− 3
> ⇔ xK
− 3
= ⇔ xK (or repeated roots)
− 3
< ⇔ xK
Circle Theorems
A tangent meets a radius at right angles
The perpendicular distance from the centre of a circle to a chord bisects that
chord
Angles in a semicircle are right angles
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Core 1 and Core 2
Curves
The angle ?gets repeated at π - ?
The angle ?gets repeated at 2π - ?
The angle ?gets repeated at π + ?
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Core 1 and Core 2
# = x;
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