Periodic Functions
Periodic Function: repeats a pattern of yvalues (outputs) at regular intervals
Cycle: may begin at any point in a graph
Period: is the horizontal length of one cycle. 
Special Right Angles
454590
h = sqrt 2 times l
306090
h = 2 times s
l = sqrt 3 times s 
s = short leg
l = long leg
Properties Of Sine Functions
y = a sin b theta
period = 2pi/b 
a = amplitude
b = number of cycles (0 to 2pi)
Quadratic Functions
Standard Form
f(x) = ax^{2} + bx + c 
ax^{2}
Quadratic term
bx
Linear term
c
constant term
Exponential Growth & Exponential Decay
b= 1 + r
b>1 = epon. growth
When b<1, b is a decay factor
xaxis = asymptote
0<b<1
b= 1+(r)
y=ab^{x} 
b= growth factor
r= increase in rate
e & Its Importance
A= amount in account
P=principal (what you start with)
r = rate in interest (annually)
t= time (in years)


Unit Circle
radian 2pi, tangent 0
radian pi/6, tangent sqrt 3/3
radian pi/4, tangent 1
radian pi/3, tangent sqrt 3
radian pi/2, tangent undefined
radian 2pi/3, tangent sqrt 3
radian 3pi/4, tangent 1
radian 5pi/6, tangent sqrt3/3
radian pi, tangent 0
radian 7pi/6, tangent sqrt3/3
radian 5pi/4, tangent 1
radian 4pi/3, tangent sqrt3
radian 3pi/2, tangent undefined
radian 5pi/3, tangent sqrt3
radian 7pi/4, tangent 1
radian 11pi/6, tangent sqrt3/3
Sine, Cosine, Tangent
Sine = opp./adj.
Cosine = Adj./Hypo.
Tangent = Opp./Adj. 
Mazimun & Minimum
y = ax^{2}+bx+c
AOS: = x = b/2a
1. vertex
2. c
3. another point 


Trigonometric Identities
Reciprocal Identities
csc theta = 1/sin theta
Sec theta = 1/ cos theta
Cot theta = 1/ tan theta
Tangent & Cotangent Identities
Tan theta = sin theta/ cos theta
Cot theta = cos theta/ sin theta
Pythagorean Identities
Cos^{2} theta + Sin^{2} theta = 1
1+ Tan ^{2} theta = Sec^{2} theta
1+ Cot^{2} theta = Csc^{2} theta 
Angle Identities
Angle Difference Identities
sin (AB) = sinA cosBcosA sinB
cos (AB) = cosA cosB + sinA sinB
tan (AB) = tanA  tan B/1+ tanA tanB
Angle Sum Identities
sin (A+B) = sinA cosB + cosA SinB
cos (A+B) = cosA cosB  sinA sinB
tan (A+B) = tanA + tan B/1tanA tanB 
Identities
DoubleAngle Identities
cos2 x = cos^{2} x sin^{2} x
cos2 x = 2cos^{2} x1
cos2 x = 1 2sin^{2}x
sin2 x = 2sin x cos x
tan2 x = 2tan x/1tan^{2}x
Half Angle Identities
sin A/2 = +/ sqrt 1cosA/2
cos A/2 = +/ sqrt 1+cosA/2
tan A/2 = +/ sqrt 1cosA/1+cosA 


Logarithms
 to base b of a positive number y is defined as...
If y=ab^{x}, then logb y= x
Log In Life
pH= log[H^{+}] 
b is not equal to 1
b must be positive
log of 0 or negative number = undefined
log= log base 10
Log Are Inverses Of Exponentials
1. Graph exponenetial function
2. Graph y = x
3. Reflect exponential function over y = x (reverse coodinates) 
Properties Of Log
logb MN = logb M+ logb N <product property
logb M/N= logb M  logb N < Quotient property
logb M^{x} = x logb M <Power property 
WATCH OUT FOR ERRORS
logb a/logb c does not equal logb a/c
logb a times c does not equal logb a times logb c
Expanding Log
log2 7b = log2 7 + log 2 b 
left to right = expand
right to left = simplify
Natural Log
Write 3ln6  ln8 as a single natural log
ln 6^{3}/8 > ln 216/8 > ln 27 
Solving Log Equations
Pt 1
solve log(3x+1) = 5
3x+1 = 10^{5}
3x+1 = 100000
3x = 99,999
x = 33,333
Pt 2
Solve 2log x log 3 = 2
log(x^{2}/3)=2
x^{2}/3 = 10^2
x^{2}= 2(100)
x=10sqrt3 or 17.32 
Pairs Of Relations are Inverse Of Each Other
y = x  7/2
y = 2x+7
y = 3x  1
y = x +1/3
y = x + 4
y = x + 4/1
y = x + 4/5
y = 5x  4 

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