Algebra and Trigonometry

Interval notation: Closed interval [ ] { }bxaxba ≤≤= |,Open interval ( ) { }bxaxba <<= |,

Half-open interval { }bxaxba <≤= |),[

Half-open interval { }bxaxba ≤<= |],(

Set notation: If A is a set of objects, we write Ax∈ if x is an object belonging to the setA. If A and B are sets, the union of A and B is { }BxAxxBA ∈∈=∪ or | .The intersection of A and B is { }BxAxxBA ∈∈=∩ and | .

BA∪ BA∩

Absolute value: The absolute value of a real number x is defined by

<−

≥=

0 if

0 if

xx

xxx .

Rules for absolute values: 1. || || aa =− 2. || || || baab =

3. ||

||

b

a

b

a= 4. |||| || baba +≤+

Exponents: For a positive integer n, 4434421 times

...n

n aaaaa ⋅⋅⋅⋅= and nn aa =1

.

If 0≠a , n

n

aa

1=− and 10 =a .

Rules for exponents: 1. mnmn aaa +=⋅ 2. mnm

n

aa

a −= 3. m nmn

aa =

4. ( ) nnn baab = 5. n

nn

b

a

b

a=

Rules for radicals: 1. nnn baab = 2. n

n

n

b

a

b

a=

Logarithms: If 0>a and xa y = , we say that yxa =log .

Rules for logarithms: 1. zxxz aaa loglog)(log += 2. zxz

xaaa logloglog −=

3. xzx az

a loglog =

Conjugates: The conjugate of an expression cba + containing a radical is cba − .

Multiplying an expression by its conjugate eliminates the radical:cbacbacba 22))(( −=−+ .

Polynomials: A polynomial of degree n in x has the form 011

1 ... axaxaxa nn

nn ++++ −

− .

Factors of a polynomial are two or more polynomials of lower degree which multiplytogether to yield the given polynomial.

Rules for factoring: 1. ))((22 bababa +−=−

2. 222 )(2 bababa +=++ 3. 222 )(2 bababa −=+−

4. ))(( 2233 babababa ++−=− 5. ))(( 2233 babababa +−+=+

Roots: A number r is a root of a polynomial 011

1 ... axaxaxa nn

nn ++++ −

− provided that

0... 011

1 =++++ −− ararara nn

nn . An thn degree polynomial can have at most n roots.

Also, if a number r is a root of a polynomial, then rx − is a factor of the polynomial.The roots of a quadratic polynomial cbxax ++2 can always be found from the

quadratic formula: a

acbbr

2

42 −±−= .

Rules for fractions: 1. BD

AC

D

C

B

A=⋅ 2.

BC

AD

D

C

B

A

DCBA

=÷=

3. Q

RP

Q

R

Q

P +=+ 4.

Q

RP

Q

R

Q

P −=−

5. BD

BCAD

D

C

B

A +=+ 6.

BD

BCAD

D

C

B

A −=−

Rules for inequalities: Let a, b, and c be any real numbers.1. If ba < and cb < , then ca < . 3. If ba < and c is positive, then bcac < .2. If ba < , then cbca +<+ . 4. If ba < and c is negative, then bcac > .

Inequalities and absolute values: Suppose a is a real number and f(x) is an expressioninvolving x.1. If axf <)( , then axfa <<− )( . 2. If axf >)( , then axf >)( or .)( axf −<

Functions: A function f is a rule that assigns to each element x in a set A one and onlyone element y in a set B, with notation )(xfy = . The set A is the domain of f, and therange of f is the set } somefor )(|{ AxxfyBy ∈=∈ .

Lines: A vertical line has an equation of the form kx = for some constant k. A line thatis not vertical must contain points ),( 111 yxP and ),( 222 yxP with 21 xx ≠ ; the slope of this

line is the number 12

12

xx

yym

−

−= . A horizontal line has slope 0. Two lines with slopes 1m

and 2m are parallel if they have the same slope ( 2m = 1m ); the lines are perpendicular if

their slopes are negative reciprocals (1

2

1

mm −= ).

Equations for lines: 1. If a line has slope m and ),( 111 yxP is a point on the line, the

point-slope form of the equation of the line is ).( 11 xxmyy −=−2. If a line has slope m and y-intercept b (i.e., it crosses the y-axis at y = b), the

slope-intercept form of the equation of the line is bmxy += .

3. If ),( 111 yxP and ),( 222 yxP are points on the line, the two-point form of the

equation of the line is 12

12

1

1

xx

yy

xx

yy

−

−=

−

−.

Circles, parabolas, ellipses and hyperbolas: A circle is the set of all points ),( yxP thatare a fixed distance r from a fixed point C(h,k). The equation of a circle with radius rand center C(h,k) is 222 )()( rkyhx =−+− .

A parabola is the set of all points ),( yxP that are equidistant from a fixed point F, calledthe focus, and a fixed line l , called the directrix. The point halfway between the focusand the directrix is the vertex of the parabola.

An equation for a parabola with vertex (h,k) and a horizontal directrix is)(4)( 2 kyphx −=− . (Parabola opens up or down)

A parabola with vertex (h,k) and a vertical directrix has the equation)(4)( 2 hxpky −=− . (Parabola opens right or left)

The sign on the constant p determines which way the parabola opens.

An ellipse is the set of all points ),( yxP with the property that the sum of the distances

from ),( yxP to the fixed points 1F and 2F is a fixed constant value. The points 1F and

2F are the foci of the ellipse, and the point midway between the foci is the center. Anequation for an ellipse with center ),( kh is

1)()(

2

2

2

2

=−

+−

b

ky

a

hx.

A hyperbola is the set of all points ),( yxP with the property that the difference of the

distances from ),( yxP to the foci 1F and 2F is a fixed constant value. A hyperbola withcenter (h,k) has equation

1)()(

2

2

2

2

=−

−−

b

ky

a

hx or 1

)()(2

2

2

2

=−

−−

a

hx

b

ky

Trigonometry: Trigonometric functions can be defined from an acute angle of a righttriangle or a central angle of the unit circle. Angles may be expressed in terms of degreesor radians. The conversion formula for changing from one to the other is

°= 180radians π .

(x,y) hyp opp θ

θ 1 adj

hyp

opp=θsin

opp

hyp==

θθ

sin

1csc y=θsin

y

1csc =θ

hyp

adj=θcos

adj

hyp==

θθ

cos

1sec x=θcos

x

1sec =θ

adj

opp==

θθ

θcos

sintan

opp

adj==

θθ

tan

1cot

x

y=θtan

y

x=θcot

Trigonometric functions of important angles:

undefined

undefined

radians

0123270

010180

01290

321

23

360

122

22

445

31

23

21

630

01000

tancossin

−°

−°

°

°

°

°

°

ππ

π

π

π

π

θθθθ

Trigonometric identities: The following equations are relationships that hold for alltrigonometric functions:

1cossin 22 =+ θθ θθ 22 sec1tan =+ θθ 22 csccot1 =+βαβαβα sincoscossin)sin( +=+ βαβαβα sinsincoscos)cos( −=+βαβαβα sincoscossin)sin( −=− βαβαβα sinsincoscos)cos( +=−

θθθ cossin22sin = θθθθθ 2222 sin211cos2sincos2cos −=−=−=

2

2cos1cos2

θθ

+=

2

2cos1sin2

θθ

−= θθ sin)sin( −=− θθ cos)cos( =−

Two useful properties related to trigonometric functions are the Law of Cosines and theLaw of Sines.

The familiar Pythagorean Theorem says that if A, B, and C are the sides of a right triangleand C is the side opposite the right angle, then 222 BAC += . The Law of Cosines is ageneralization of the Pythagorean Theorem. If, as shown in the triangle below, θ is theangle opposite side C, then the Law of Cosines states that

A C θcos2222 ABBAC −+=

θ

B

The Law of Sines gives a relationship between the three sides and the three angles of ageneral triangle. If α is the angle opposite side A, β is the angle opposite side B, and θ isthe angle opposite side C, then the Law of Sines states that

β

A Cθβα sinsinsin

CBA==

θ α

B