Area of a Triangle
A = 1 ab sin C
2
where a and b are sides of a triangle
where C is the angle between those sides
must have SAS to use formula
C
b
A
Example:
a
c
B
Find the exact value of the area of an equilateral triangle if the
measure of one side is
Laws of Exponents
When multiplying the exponents of like bases, add the
exponents and keep the bases the same.
xa xb = xa + b
Example:
When dividing the exponents of like bases, subtract the
exponents and keep the bases the same.
xa
= xa b
b
x
Example:
(x
Law of Cosines
1.
Law of Cosines is used to find missing pieces of a triangle.
2.
Law of Cosines is used only when you know:
a)
b)
3.
SAS, or
SSS
The triangle does not have to be a right triangle.
Law of Cosines General Equation
C
b
A
a
c
B
c 2 = a 2 + b
Exponential Functions
Definition of an Exponential Function:
f (x ) = b x , where b > 0, b 1
Example:
This means that the exponent is a
variable, the base is positive, but not
equal to one.
Which of the following is an exponential functions?
(1)
(2)
y = 1
Law of Sines
1.
Law of Sines is used when the problem is not Law of Cosines.
2.
The triangle does not have to be a right triangle.
Law of Sines General Equation
c
a
b
=
=
sin A sin B sin C
The ratio of a side of a triangle over the sine of its opposite an
Sum and Product of Roots of a Quadratic Equation
Formulas for Sum and Product of Roots
sum of roots =
b
a
sum of products =
c
a
where a, b, and c are coefficients in ax2 + bx + c = 0
Example:
Find the sum and product of the roots of:
x2 + 6x + 8 = 0
Using
Solving Quadratic Equations
with Imaginary or Complex Roots
Example:
Solve. Express roots in simplest a + bi form.
x 2 8 x = 17
x 2 8 x + 17 = 0
a=1
b b 2 4ac
x=
2a
x=
( 8)
8 4
2
8 2i
x=
2
x=4i
x=
( 8)2 4(1)(17 )
2(1)
b = -8
c = 17
Example:
Solve. Expr
Logarithmic Functions
Definition of Logarithm:
A logarithm is an inverse exponential function.
The logarithm of a number is an exponent.
Exponential Functions
Logarithm
Inverse Exponential Function
f(x) = 2x
x = 2y
x
-2
-1
0
1
2
x
1
2
4
y
1
2
4
y
-2
-1
0
Equations with Fractional Exponents
Procedure:
1.
2.
3.
Get x by itself.
Make power of x 1.
Solve.
Example:
1
Solve x 2 = 4 .
1
2
x =4
12
2
x 2 1 = 41
x = 16
Example:
1
2
Solve 2 x = 18 .
1
2
2 x = 18
1
2 x 2 18
=
2
2
12
21
x
=9
x = 81
Example:
2
1
3
Solv
Imaginary Numbers
x2 = 4
x=2
Solve:
x2 = 4
x = 2i
i = 1
i1 = i
i 5 = i 4 i1 = i
i2 = 1 1 = 1
i6 = i4 i2 = 1
i 3 = i 2 i1 = i
i7 = i4 i3 = i
i4 = i2 i2 = 1
i8 = i 4 i 4 = 1
With the imaginary number system, never multiply negative radicands.
You must simpl
Logarithmic Form of an Equation
Exponential Form
Logarithmic Form
x = 2y
where 2 is a base and y is an
exponent
y = log 2 x
where 2 is a base and y is an
exponent
Example:
Write 8 2 = 64 in logarithmic form.
8 2 = 64
2 = log 8 64
Example:
Write log 3 81 =
Laws of Logarithms
1.
2.
3.
log b xy = log b x + log b y
x
log b = log b x log b y
y
When multiplying, add.
When dividing, subtract.
log b (x ) = p log b x
Example:
When power-to-power, multiply.
p
Rewrite the following expressions using the Laws of
Logar
Scientific Notation
a 10 n
where a is a number between 1 and 10
where n is an integer
Regular Numbers
2,854
285.4
28.54
2.854
.2854
.02854
Example:
Scientific Notation
2.854 x 103
2.854 x 102
2.854 x 101
2.854 x 100
2.854 x 10-1
2.854 x 10-2
Express
(7.5