159258080-CXC-MATHS-Formula-Sheet.doc

Formula Sheet 4 CXC Trigonometric Formula

Area and Perimeter Formula Perimeter = distance around the outside (add all sides).

Opposite – side opposite to angle Adjacent –side beside (adjacent) to angle

hyp θ 

Hypotenuse- longest side opposite sin hypotenus cos tan

c

adj  a

adjacent 

hyp

opp  b

hypotenus opposite

Remember: or!s only on rig"t angle triangles

adjacent 

Pyt"agorean T"eorem Triples: Tr iples: #$%$&

&$'$'#

$'&$'*

+ is t"e "ypotenuse$ a and b are t"e ot"er sides . Remember: or!s only on rig"t angle triangles

+oordinate ,eometry Formula 2olume and 3ur4ace Area

istance Formula:

.idpoint Formula:

,radient Formula:

50uation o4 a line Slope-Intercept Method:

 y

mx

c

Point-Gradient Method:

Parallel lines "a/e e0ual slope1 Perpendicular lines "a/e negati/e reciprocal gradients1

Parallel lines

Angle Information Complementary angles - to angles hose sum is !". Supplementary angles - to angles hose sum is #$".

Corresponding angles are e%ual. =

General Triangle Information

'

*=

+'

,=

Sum o1 angles o1 triangle = #$".

lternate Interior angles are e%ual.

Measure o1 e/terior angle o1 triangle = the sum o1 the to non-ad2acent interior angles.

'

3he sum o1 any to sides o1 a triangle is greater than the third side.

lternate /terior angles are e%ual. #= $' = +

,=

#=

&'

$ *=

&

Same side interior angles are supplementary. m *0m &=#$"' m ,0m =#$"

3ol/ing triangles Polygons

c

b

Sum o1 Interior ngles:  B a

Sum o1 /terior ngles:

ach Interior ngle (regular poly):

ach /terior ngle (regular poly):

6uadratic Formula

C 

3ine rule a

b

c

 or

sin  A

sin B

sin C 

a b c sin A sin B sin C  4sed hen any to sides and their corresponding angles are in5ol5ed to 1ind one missing side or angle.

+osine rule a

b

c

bc cos A

b

a

c

ac cos B

c

a

b

ab cos C

 

84 

ax 

  t"en

bx

c

4sed hen three sides and an angle beteen them are gi5en to 1ind the other side Heron7s Formula Area o4 a triangle gi/en only t"e lengt" o4 t"e sides

9 b

b

 x 

%ac

 A

s( s

a where s

a )( s a

b

b )( s

c)

c

 +apital letters represent Angles +ommon letters represent sides

Tangent

+ircle Facts

&1 Radius to tangent is 9 o at point o4 contact1 ;1 T"e tangents to a circle 4rom an e
radius

*1 Angle beteen tangent to circle and c"ord at t"e point o4 contact is e0aual to t"e to t"e angle in t"e alternet segment

diameter

1

segment 'a.

6 3ector

%1

iameter = > radius

#1

Area o4 circle = ?r +ircum4erence o4 circle = ?r or ?d @engt" o4 arc = Area o4 sector =

?r >

#;9

;1  &1



?r  >

#;9

 Area of   Segment  =  Area of   sec tor  −  Area of   triangle θ     # =    π r  ×   −  r  *)"      ( (

(

sin θ 

     

3

B

+.

Angles in circles

A E D

T

'1

a$b$c$d Angle at t"e center in tice

#b.

#c.

angle at t"e circum4erence1 1

Angle 4ormed on t"e diameter in 9 o

#1

Angles in t"e same segment are e0ual

#d. %1

opposite angles in a cyclic 6uadrilateral are supplimentary( add up to '9o)

Trans4ormational .atrices

.atrices  Adding or subtracting matrices a b e f a e b c

d

g

h

c

 Multiplying Matrices a b e f ae c

d

g

h

ce

g

f

d

 

h

ad

 x * axis ,

bg

af

bh

dg

cf

dh

 Determinant of 2×2 Matrix  a b  A c d   f  A

 #$%&$C'()  Multiply matrices by each point to get reflection in

!

 y* axis

!

,

,

!

y+x

! ,

y+*x 

!

,

,

!

! ,

, !

'#A)-&A'() 

 Mo"ement of x in x direction and y in ydirection add matrix   x  to eac. point to get its image/  y

 #('A'()   Multiply by matrices to get rotation of ϴ *degrees cloc.wise about origin 0!1!

cb

 A singular matrices has a determinant of !  Adjoint of 2×2 matrix  a b  A c d   A adjo int

d c

 #

cos sin

b a

sin

 #,4!

 #3!

cos

, !

!

 #25!

,

!

,

,

!

! ,

, !

 $nlargment 

 n"erse of 2×2 matrix 

 Multiply each point by scale factor 6 to get the image of the  point for an enlargment from the origin/

 -ets

 A '

 A

a

b

uni/ersal set

c

d 

is a member o4 

'

is not a member o4 

A  adjoint

 A

union intersect

or   A

d

'

'

ad

bc

c

= null set 7   A + 5lements not in set A

b a

 #e"erse oppertions

 A 

 x 

a

a

 x 

sin

sin

'

cos

cos

'

tan

tan

'

 f ( x )

f

'

( x )

Shape

7olume

Sur1ace rea

Cube 3

l×l×l=l 

2

6l 

1 l  l 

Cuboid lwh

2lw+2hw+2lh

h l  w

Prism sl  shl 

bh lb h

,

bhl 

2

l  b

Cylinder  r  2

2 r 

2

2 r  h

rh

h or  2 r 0 r 

h 

Cone ,

2

r 

8

r 2 hrs

Sphere 9

r 8 8 9 r 2

8"" 9. erguson