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Exponents and Radicals

b n = b b b ... b , where

The nth power of b

14243

n factors of b

b is the base,

n is the exponent,

b n = a

n a = b :

The nth root of a (radical)

a is called the radicand, n is called the index

of a root.

a square root of a

2 a = a

the cube root of a;

if n = 2k + 1 (n is odd),

• n a n =

, if n = 2k (n is even).

Remember:

• b 0 = 1, b ≠ 0

• b −n =

and

= b n . b ≠ 0 and

n ≠ 0 ,

b n

−n

= b1/ n , n is a positive integer and b is a

• n

real number

Properties of Exponents

Product

• b m b n = b m+n

• (ab)n

= a n b n

Quotient

= b m−n

•

b n

Exponent

• (b m )n = b mn

Properties of Radicals

Product

n a n b = n ab

Quotient

(b ≠ 0)

= n

n b

Radical

m n

Exponent

)n = n

b n

= b

Fundamental identity

km b kn = m b n

Polynomials

The general form of a polynomial

Pn (x) = an x n + an−1 x n−1 + L + a2 x 2 + a1 x + a0

leading coefficient

a n

constant term

• P0 (x) = a0

Particular cases of polynomials

• P1 (x) = a1 x + a0

• P (x)

= a

x 2

+ a x + a

………………………….

Sum (difference) of polynomials

combine like terms

multiply every monomial of the first

Product of polynomials

polynomial by every monomial of the second

one and combine like terms.

Special Product Formulas

Difference of the squares

x 2 − y 2 = (x − y ) (x + y )

Sum (difference) of the cubes

x 3 ± y 3 = (x ± y ) (x 2 m xy + y 2 )

Square of the sum (difference)

(x ± y )2 = x 2 ± 2xy + y 2

Cube of the sum (difference)

(x ± y)3 = x3 ± 3x2 y + 3xy2 ± y3 =

= x3 ± y3 ± 3xy(x ± y)

Rational fraction (algebraic fraction)

Pn (x)

Qm (x)

Pn (x)

− proper rational fraction

Qm (x)

n < m

Pn (x)

− improper rational fraction

n ≥ m

Qm (x)

Logarithms

Logarithm of x to the base a (logarithm is

log a x (log a x = n a n = x)

an exponent of a base a to receive

argument

Properties of Logarithms

Fundamental identities

• logb b = 1

• logb 1 = 0

An inverse property	logb (b			p		)= p
	logb (b					)= p
	b	logb	p	= p (for p > 0)
	b			= p (for p > 0)

Product property	log b MN = log b M + log b N
Quotient property	logb		M		= log b M – log b N
	logb				= log b M – log b N
			N
Power property	log b (M p )= p log b M
One-to-one property	log b M = log b N M = N
Logarithm of each side property	M = N log b M = log b N
Change-of-Base Formula	logb		x =				log a x
a, x, and b are positive real numbers with	logb		x =				log a b
a ≠ 1 and b ≠ 1,							log a b
a ≠ 1 and b ≠ 1,
common logarithm ( log10 x = lg x )	lg x
natural logarithm ln x = (log e x)	ln x

Trigonometric Functions of an Acute Angle

Let α be an acute angle of a right triangle. The values of four trigonometric functions of α are

sin α =

length of opposite side

length of hypotenuse

cosα =

length of adjacent side

length of hypotenuse

tan α = length of opposite side = y

length of adjacent side

cot α =

length of adjacent side

length of opposite side

= π / 6

= π / 4

= π / 3

sin α

1/2

1 /

2 =

2 / 2

3 / 2

cosα

1/2

3 / 2

1 /

2 =

2 / 2

tan α

1 /

3 = 3 / 3

cot α

1 /

3 =

3 / 3

	y			In general
D	O2			In general
D	O2			sin α = y		, cosα = x ,
	A			sin α = y		, cosα = x ,
	A				r				r
					r				r
	α		x	tan α =	y	, sin	2	x	+ cos	2	x = 1,
	α		x				2			2
B	O	O1			x
B	O	O1			x
				where r = OP =					x 2 + y 2 .
		C

Trigonometric Circle

Let the radius of circle be equal to 1, then

the vertical diameter is called the sines-line sin α = y = OA the horizontal diameter – the cosines-line cosα = x = OB the upper tangent – line of cotangent tan α = O1C

the right tangent – line of tangent cot α = O2 D

Trigonometric Formulas

The Fundamental trigonometric

• sin 2 x + cos 2 x = 1

Identities

• tan x =

sin x

• cot x =

cos x

sin x

• 1 + tan 2 x =

• 1 + cot 2

x =

cos 2 x

sin

2 x

Sum and difference Identities

sin(α ± β ) = sin α cos β ± cosα sin β

tan(α ± β ) =

tan α ± tan β

1 m tan α tan β

cos(α ± β )= cosα cos β m sin α sin β

cos 2x = cos 2 x − sin 2 x

Double-Angle Identities

sin 2x = 2 sin x cos x

tan 2x =

2 tan x

1 − tan 2 x

sin

= ±

1 − cos x

Half-Angle Identities

cos

= ±

1 + cos x

tan

= ±

1 − cos x

1 + cos x

sin(900 − x)= cos x

Co-function Identities

cos(900 − x)= sin x

tan(900 − x)= cot x

cot(900 − x)= tan x

sin x cos x =

sin 2x

Lowering of the Order

cos 2 x =

+ cos 2x

sin 2 x =

1 − cos 2x

cosα cos β =

(cos(α + β ) + cos(α − β ))

Product-to-Sum Identities

sin α sin β = −

(cos(α + β ) − cos(α − β ))

cosα sin β =

(sin(α + β ) − sin(α − β ))

sin α cos β =

(sin(α + β ) + sin(α − β ))

cos x + cos y = 2 cos

x + y

cos

x − y

cos x − cos y = −2 sin

x + y

sin

x − y

Sum-to-Product Identities

x + y

x − y

sin x + sin y = 2 sin

cos

sin x − sin y = 2 sin

x − y

cos

x + y

a sin x + b cos x = k sin(x + α ), where

k = a 2 + b 2 , sin α =

cosα =

a 2

+ b 2

a 2 + b 2

Equations

Linear equation

ax + b = 0

x = −

Quadratic equations:

− b ±

b 2

− 4ac

1,2

ax 2 + bx + c = 0

+ x2 = −

x x

ax 2 + 2kx + c = 0

x1,2

− k ±

k 2

− ac

Vieta’s theorem: if x1 and x2 are roots of

+ x

= − p,

an equation x 2 + px + q = 0 , then

x1 x2 = q.

Exponential equation a f (x )

= b ,

a > 0, a ≠ 1

f (x) = log a b

Logarithmic equation log a

f (x) = b ,

f (x) = a b

a > 0, a ≠ 1

f (x) > 0

Trigonometric equations

sin f (x) = a,

≤ 1

f (x) = (− 1 )n arcsin a + nπ , n Z

cos f (x) = a,

≤ 1

f (x) = ± arccos a + 2nπ , n Z

tan f (x) = a

f (x) = arctan a + nπ , n Z

cot f (x) = a

f (x) = arc cot a + nπ , n Z

≤ 1, x [− π / 2,π / 2]

• arcsin a = x :

sin x = a, where

• arccos a = x :

cos x = a, where

≤ 1, x [0,π ]

• arctan a = x :

tan x = a, where

x (− π / 2,π / 2)

• arc cot a = x :

cot x = a,

x ( 0,π )

COMPLEX NUMBERS

Vocabulary

complex number – комплексное число

conjugate complex number – сопряженное комплексное число exponential form – показательная форма

geometrical representation – геометрическая интерпритация imaginary axis – мнимая ось

imaginary part – мнимая часть imaginary unit – мнимая единица

integral powers and roots of complex numbers – целые степени и корни комплексных чисел

principal argument – главный аргумент real axis – действительная ось

real part – действительная часть quadrant – четверть, квадрант

• By a complex number we mean an ordered pair of real numbers z= (x, y) where x – real part, y – imaginary part

•The equality relation and the arithmetical operations:

1)(x1, y1 ) = (x2 , y2 ) x1 = x2 , y1 = y 2 ;

2)(x1 , y1 )± (x2 , y2 ) = (x1 ± x2 , y1 ± y 2 );

3)(x1 , y1 ) (x2 , y2 ) (x1 x2 − y1 y2 , x1 y2 + y1 x2 ).

• Imaginary unit: i = (0,1 );

		1,	if n = 4k

• i	n	i, if n = 4k + 1
• i		=	, where k N
		− 1, if n = 4k + 2

− i, if n = 4k + 3

• Standard or rectangular form: z = x + yi

					y
• Modulus, or absolute value:	z	= ρ =	x 2 + y	2 .	y	z - ( x,y)
					y
• The conjugate to z = x + yi is x − yi and is denoted by z :							x
z = x − yi					0
z = x − yi						-
					-y	-	- (x,-y)
					-y	z	- (x,-y)
• Division of complex numbers:

if z		≠ 0 then z =	z1	=	z1	z2	.
	2		z 2

					z 2 z2

y
	ρ
		y
	ϕ	x
0	x

If the point	z = (x, y) = x + yi is represented by polar
coordinates ρ and ϕ ,we can write
x = ρ cosϕ
	then
y = ρ sin ϕ

•z = ρ (cosϕ + i sinϕ ) – is the trigonometric form of a complex number .

•The x-axis is called the real axis.

•The y-axis is the imaginary axis.

• The unique real number ϕ which satisfies the condition − π < ϕ ≤ π is called the principal argument of z and is denoted by arg z : ϕ = arg z and can be calculated by

	y
arctan		, if z I or IYquadrant,
arctan		, if z I or IYquadrant,
	x
formulas: ϕ = arg z =			y
π + arctan			y	, if z II or III quadrant.
π + arctan				, if z II or III quadrant.
			x

•z = ρ e iϕ ( z = ρ exp(iϕ )) is an exponential form of a complex number .

•Sometimes there is used the next form of complex number: z = ρ ϕ .

Let z1 = ρ1 ϕ1 and z 2 = ρ 2 ϕ 2 be two complex numbers, then

Operation

Trigonometric form

Exponential form

Product

z1 z 2 =

= ρ1 ρ 2 (cos(ϕ1 + ϕ 2 ) + i sin(ϕ1 + ϕ 2 ))

i (ϕ1+ϕ2 )

ρ1 ρ 2 e

Ratio

ρ1

(cos(ϕ1 − ϕ2 )+ i sin(ϕ1

− ϕ2 ))

ρ1

i (ϕ1−ϕ2 )

ρ 2

Integral

power z n =

= ρ n (cos nϕ + i sin nϕ )

= ρ n einϕ

ϕ + 2kπ

Integral root

= n

ρ cos

+ i sin

= n

ρ exp i

n z = wk

k = 0,1,2,..., n − 1.

Remember :

•e −iϕ = cos ϕ − i sin ϕ

•cos ϕ = eiϕ + e−iϕ

• sin ϕ = eiϕ − e−iϕ . 2i

FUNCTIONS

Vocabulary

common logarithm. (log10 x) − десятичный логарифм composite function – сложная функция

cube parabola – кубическая парабола domain – область определения

even function – четная фyнкция

exponential function – показательная фyнкция function - функция

graph of a function – график фyнкция hyperbola - гипербола

inverse function – обратная фyнкция linear function – линейная фyнкция maximum - максимум

minimum – минимум

natural exponential function (e x )− экспонента natural logarithm (ln x) − натуральный логарифм odd function – нечетная фyнкция

one-to-one function – взаимно-однозначная фyнкция parabola – парабола

periodic function – переодическая фyнкция power function – степенная фyнкция quadratic function–квадратичная фyнкция range – область значений

root or a zero of a function – корень или ноль функции slope (a = tan α ) - угловой коэффициент

straight line –прямая

y-intercept point – пересечение с осью ОУ

Names	Definitions, theorems
	Correspondence of sets D and		E : x D
Function: y = f (x)	exactly one y E

Domain of a function: D(y )	D(y ) = {x : f (x) takes finite, real values}

Roots (zeros) of f (x)	{x : f (x)= 0}- Set of x for which		f (x) = 0

	{(x; f (x))}− A	set of points of	a plane with
Graph of a function y = f (x)	coordinates x and f (x)

A function f(x) is increasing if	f (x1 ) < f (x2 )	whenever x1 < x2

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