More Related Content Similar to K to 12 math (20) K to 12 math1. ๐ฒ๐๐๐๐ ๐. ๐พ๐๐๐๐๐ ๐น๐๐๐๐ IX
Basic Algebra
Algebraic Rules and Laws of Indices
Commutative Law (๐ + ๐) = (๐ + ๐) ;(๐ โ
๐) = (๐ โ
๐)
Associative Law ๐ + (๐ + ๐) = (๐ + ๐) + ๐ ; ๐ โ
(๐ โ
๐) = (๐ โ
๐) โ
๐
Distributive Law ๐ โ
(๐ + ๐) = ๐ โ
๐ + ๐ โ
๐ ;(๐ + ๐) ๐โ = ๐ ๐โ + ๐ ๐โ
The Product and Quotient Rules ๐ โ
๐ = ๐ ; ๐ ๐โ = ๐ ;โ๐ โ
๐ = โ ๐ โ
โ๐
The First Power Distributive Rule (๐ ) = ๐ โ
;(๐ ๐โ ) = (๐ ) (๐ )โ ;(๐ โ
๐) = ๐ โ
๐
The Root and Inverse Identity ๐ โ
= โ ๐ = โ ๐ ; ๐ = 1 ๐โ ; ๐ ๐โ = โ ๐ โ๐
Algebraic Formulae
Quadratic Formula Common Functions Linear Exponential Power
๐ =
ยฑโ
๐( ๐ฅ) = ๐ฆ
๐ ๐โ1
( ๐) = ๐
๐(๐ฅ) = ๐๐ฅ + ๐ ๐(๐ฅ) = ๐ โ
๐ โ
๐(๐ฅ) = ๐ โ
๐ฅ
Logarithmic Algebra Logarithmic Applications Exponential Forms
โ๐ = ๐ โ ๐๐๐ (๐) = ๐ โ ๐ = ๐ ๐๐๐ (๐ด๐ต) โ ๐๐๐ (๐ด) + ๐๐๐ (๐ต) Base Greater than 1
โ๐ = ๐ โ ๐๐(๐) = ๐ โ ๐ = ๐ ๐๐๐ (๐ด ๐ตโ ) โ ๐๐๐ (๐ด) โ ๐๐๐ (๐ต) ๐ฆ = ๐ฆ โ
๐ยฑ
๐๐๐ ( ๐ด ) = ๐ต โ
๐๐๐ ๐ด Rate Form
๐ ๐๐
โ ๐ โ โ ๐
๐๐๐ ๐( ๐)
๐ฆ = ๐ฆ โ ๐ยฑ
๐๐๐ ๐
( ๐ ๐
) โ ๐ โ ๐๐๐ โ ๐
๐ ( ๐) Time Constant Form
โ ๐
๐ ๐
โ ๐ โ ๐๐๐๐ ๐
( ๐)
๐ฆ = ๐ฆ โ
๐ยฑ โ
๐๐(1) = 0 ๐๐(๐) = 1 Growth/Decay by Factor in time
๐ฆ = ๐ฆ โ
๐ โ
Algebraic Identities from Binomial Theorem
๐2
+ 2๐๐ + ๐2
= ( ๐ + ๐)( ๐ + ๐) = ( ๐ + ๐)2
๐3
+ 3๐2
๐ + 3๐๐2
+ ๐3
= ๐3
+ ๐3
+ 3๐๐( ๐ + ๐) = ( ๐ + ๐)3
๐2
โ 2๐๐ + ๐2
= ( ๐โ ๐)( ๐โ ๐) = ( ๐โ ๐)2
๐3
โ 3๐2
๐ + 3๐๐2
โ ๐3
= ๐3
โ ๐3
โ 3๐๐( ๐โ ๐) = ( ๐โ ๐)3
๐2
โ ๐2
= ( ๐ + ๐)( ๐ โ ๐)
๐2
+ ( ๐ด + ๐ต) ๐ + ๐ด๐ต = ( ๐ + ๐ด)( ๐ + ๐ต)
๐2
+ ๐2
+ ๐2
+ 2๐๐ + 2๐๐ + 2๐๐ = ( ๐ + ๐ + ๐)( ๐ + ๐ + ๐) = ( ๐ + ๐ + ๐)2
๐ด๐ถ๐ + (๐ด๐ท + ๐ต๐ถ)๐ + ๐ต๐ท = (๐ด๐ + ๐ต)(๐ถ๐ + ๐ท)๐ + ๐ = (๐ + ๐)(๐ โ ๐๐ + ๐ )๐ โ ๐ = (๐โ ๐)(๐ + ๐๐ + ๐ )
( ๐ + ๐) ๐
= ๐ ๐
+ ๐ โ
๐ ๐โ1
๐ฆ +
๐( ๐โ1)
2
๐ ๐โ2
๐2
+ โฏ +
๐
๐
๐ ๐โ๐
๐ ๐
+ โฏ + ๐๐๐ ๐โ1
+ ๐ ๐
๐คโ๐๐๐
๐
๐
=
๐( ๐โ1)โฆ( ๐โ๐+1)
1โ
2โ
3โฆ๐
X Z
Y
2. ๐ฒ๐๐๐๐ ๐. ๐พ๐๐๐๐๐ ๐น๐๐๐๐ XI
Basic Trigonometry
Circles Triangles
The degree 1o = 1/360 of a circle โ 0.01745 radian Pythagoras, right triangle A2+B2 = C2
The radian 1rad = 1/2ะฟ of a circle โ 57.30o Degrees of internal angles ao+bo+co = 180o
The grad 1grad = 1/400 of a circle sinฮธ = O/H cosฮธ = A/H tanฮธ = O/A
90o = right angle ะฟ/2 = right angle 100grads = right angle
180o 0o, 360 ะฟ 0, 2ะฟ 200 0, 400
270o 3ะฟ/2 300
Degrees Radians Grads
Conversions of Angles Circular Measure of degrees
TO degrees radians grads 1 degree = 60 minutes
FROM 1 degree = 3600 seconds
degrees - - - - - - โข(ะฟ/180) /0.9 1 minute = 60 seconds
radians โข(180/ะฟ) - - - - - - โข(200/ะฟ) 1 circle = 21'600 minutes
grads โข0.9 โข(ะฟ/200) - - - - - - 1 circle = 1'296'000 seconds
Cartesian Quadrants
Trig Equations
Sine Laws
( )
=
( )
=
( )
๐ด =
( )
( )
Cosine Laws Area Formula Symbolic Trigonometric Ratios
๐ด = ๐ต + ๐ถ โ 2๐ต๐ถ๐๐๐ (๐)
๐ต = ๐ด + ๐ถ โ 2๐ด๐ถ๐๐๐ (๐)
๐ถ = ๐ด + ๐ต โ 2๐ด๐ต๐๐๐ (๐)
( ) ( )
( )
=
( ) ( )
( )
=
( ) ( )
( )
Trigonometric Identities
๐ ๐๐(โ๐ผ) = โ๐ ๐๐(๐ผ) ๐๐๐ (โ๐ผ) = ๐๐๐ (๐ผ) ๐ก๐๐(โ๐ผ) = โ๐ก๐๐(๐ผ)
๐๐๐ 2๐ = 1 โ (2๐ ๐๐ ๐) ๐๐๐ 2๐ = (2๐๐๐ ๐) โ 1 tan(๐ผ) =
( )
( )
๐ ๐๐(180ยฐ โ ๐) = ๐ ๐๐(๐) ๐ ๐๐(90ยฐ โ ๐) = ๐๐๐ (๐) ๐ ๐๐2๐ = 2๐ ๐๐๐๐๐๐ ๐
๐๐๐ (180ยฐ โ ๐) = โ๐๐๐ (๐) ๐๐๐ (90ยฐ โ ๐) = ๐ ๐๐(๐)
Product Relations Pythagorean Relations
๐ ๐๐๐ด = ๐ก๐๐๐ด๐๐๐ ๐ด ๐๐๐ ๐ด = ๐๐๐ก๐ด๐ ๐๐๐ด ๐ก๐๐๐ด = ๐ ๐๐๐ด๐ ๐๐๐ด ๐ ๐๐ ๐ด + ๐๐๐ ๐ด = 1 1 + ๐ก๐๐ ๐ด = ๐ ๐๐ ๐ด
๐๐๐ก๐ด = ๐๐๐ ๐ด๐๐ ๐๐ด ๐ ๐๐๐ด = ๐๐ ๐๐ด๐ก๐๐๐ด ๐๐ ๐๐ด = ๐ ๐๐๐ด๐๐๐ก๐ด 1 + ๐๐๐ก ๐ด = ๐๐ ๐ ๐ด ๐๐๐ 2๐ = ๐๐๐ ๐ โ ๐ ๐๐ ๐
Angle-sum, Angle-differences relations and Double-angle Relations
๐ ๐๐(๐ด + ๐ต)๐ ๐๐(๐ด โ ๐ต) = ๐ ๐๐ ๐ด โ ๐ ๐๐ ๐ต = ๐๐๐ ๐ต โ ๐๐๐ ๐ด ๐ ๐๐(๐ด ยฑ ๐ต) = ๐ ๐๐๐ด๐๐๐ ๐ต ยฑ ๐๐๐ ๐ด๐ ๐๐๐ต
๐๐๐ (๐ด + ๐ต)๐๐๐ (๐ด โ ๐ต) = ๐๐๐ ๐ด โ ๐ ๐๐ ๐ต = ๐๐๐ ๐ต โ ๐ ๐๐ ๐ด ๐๐๐ (๐ด ยฑ ๐ต) = ๐๐๐ ๐ด๐๐๐ ๐ต โ ๐ ๐๐๐ด๐ ๐๐๐ต
๐ก๐๐(๐ด ยฑ ๐ต) =
ยฑ
โ
๐๐๐ก(๐ด ยฑ ๐ต) =
( ) ( )ยฑ
( )ยฑ ( )
๐๐๐ก2๐ด = ๐ก๐๐2๐ด =
๐๐๐ 2๐ด = ๐๐๐ ๐ด โ ๐ ๐๐ ๐ด = 2๐๐๐ ๐ด โ 1 = 1 โ 2๐ ๐๐ ๐ด = ๐ ๐๐2๐ด = 2๐ ๐๐๐ด๐๐๐ ๐ด =
Function-product Relations, Function-sum and Function-difference Relations
๐ ๐๐๐ด๐ ๐๐๐ต =
( )
โ
( )
๐ ๐๐๐ด๐๐๐ ๐ต =
( )
+
( )
๐ ๐๐๐ด ยฑ ๐ ๐๐๐ต = 2๐ ๐๐
ยฑ
๐๐๐
โ
๐๐๐ ๐ด๐ ๐๐๐ต =
( )
โ
( )
๐๐๐ ๐ด๐๐๐ ๐ต =
( )
+
( )
๐๐๐ ๐ด + ๐๐๐ ๐ต = 2๐๐๐ ๐๐๐ ๐๐๐ ๐ด โ ๐๐๐ ๐ต = โ2๐ ๐๐ ๐ ๐๐ ๐ก๐๐๐ด ยฑ ๐ก๐๐๐ต =
( ยฑ )
Trig Definitions
sinฮธ = opposite/hypotenuse = O/H = y/r = 1/csc Arcsinฮธ = sin-1
cosฮธ = adjacent/hypotenuse = A/H = x/r = 1/sec Arccosฮธ = cos-1
tanฮธ = opposite/adjacent = O/A = y/x = 1/cot Arctanฮธ = tan-1
cscฮธ = hypotenuse/opposite = H/O = r/y = 1/sin Arccscฮธ = csc-1
secฮธ = hypotenuse/adjacent = H/A = r/x = 1/cos Arcsecฮธ = sec-1
cotฮธ = adjacent/opposite = A/O = x/y = 1/tan Arccotฮธ = cot-1
A B
C (C โ y)y
x
bo ao
co
2nd
+sin +csc
-cos -sec
-tan -cot
1st
+sin +csc
+cos +sec
+tan +cot
-sin -csc
+cos +sec
-tan -cot
4th
-sin -csc
-cos -sec
+tan +cot
3rd
x
y
ฮธ
Sinฮธ
Cos ฮธ
Cot ฮธ
Csc ฮธ Tan ฮธ
Sec ฮธ