Section 4.1 The Law of Sines
An oblique triangle is a triangle that does not contain a right angle.
Consider the following triangles.
C
b
A
C
γ
α
h
β
c
A is acute
γ
a
h
B
a
b
α
A
β
c
B
A is obtuse
To solve an oblique triangle, you need to know the measure of at least one side and any
two other parts of the triangle. This breaks down into four cases.
1.
2.
3.
4.
Two angles and any side (AAS or ASA).
Two sides and an angle opposite one of them (SSA).
Three sides (SSS).
Two sides and their included angle (SAS).
The first two cases can be solved using the Law of Sines, whereas the last two cases
require the Law of Cosines.
Law of Sines
If ABC is a triangle with sides a, b, and c, with height h, then
sin   sin   sin  


.
a
b
c
In words, the law of sines states the following. In any triangle, the ratio of the sine of an
angle to the side opposite that angle is equal to the ratio of the sine of another angle to the
side opposite that angle.
When you are trying to solve an oblique triangle where you are given two sides and one
opposite angle (SSA) you may run into difficulty. The reason is that the information
given does not guarantee a unique triangle or that a triangle even exists with the given
information.
1
Consider the pictures above under the law of sines. The following conditions will
determine how many solutions are possible.
Type of Angle
α is Acute
α is Obtuse
Necessary Condition
ah
ah
a b
hab
ab
a b
Number of Solutions
None
One
One
Two
None
One
In practical terms, the information in the table above can be interpreted in the following
way.



If sin   1 , no triangle exists.
If sin   1 , then   90  and we can solve the triangle as before. In this case,
only one triangle exists.
If sin   1 , then zero, one, or two triangles will form (called the ambiguous
case).
o If   90  , then two triangles form
o If   90  and a  b , then only one triangle will form.
Homework: 1-15 odd.
2
Section 4.2 The Law of Cosines
Law of Cosines
If ABC is a triangle with sides a, b, and c, with height h, then.
a 2  b 2  c 2  2bc cos 
b 2  a 2  c 2  2ac cos 
c 2  a 2  b 2  2ab cos 
In words, the law of cosines states the following. The square of the length of any side of a
triangle equals the sum of the squares of the other two sides minus twice the product of
the lengths of the other two sides and the cosine of the angle between them.
The law of cosines is used when
1. We know two sides and the angle between them (SAS), or
2. We know the lengths of all three sides of the triangle (SSS).
Hints:
When you have the condition of SAS, use the law of cosines to find the other side; then
use the law of sines to find the angle opposite the shortest side. This avoids the issues of
the ambiguous case.
When you have the condition of SSS, find the largest angle first, which is the angle
opposite the largest side. Then use the law of sines or cosines to find one of the remaining
angles.
The Area of a Triangle
There are three general formulas to find the area of an oblique triangle.
1
A  bc sin 
2
1
A  ac sin 
2
1
A  ab sin 
2
In words, these formulas state the following. The area of a triangle equals one-half the
product of the lengths of any two sides and the sine of the angle between them.
3
There is an additional formula one can use to find the area of an oblique triangle if you
only know the sides of the triangle.
Heron’s Formula
The area of a triangle with sides a, b, and c is given by A  ss  a s  bs  c  where s
1
is one-half the perimeter; that is s  a  b  c  .
2
Homework: 1-9 odd, 29-35 odd.
4
Section 4.3 Trigonometric Form for Complex Numbers
A complex number is a number that can be represented in the form a  bi , where a and
b are real numbers and bi is an imaginary number.
We can geometrically represent (i.e. graph) a complex number a  bi in a plane using
the ordered pair (a, b). The x-axis is relabeled as the real axis and the y-axis is relabeled
as the imaginary axis.
Imaginary axis
(a+bi)
r
b
θ
a
Real axis
The absolute value of a complex number z  a  bi is defined to be the distance the
point (a, b) is from the origin when graphed in a plane. As a result, if z  a  bi , then
z  a  bi  a 2  b 2 .
The figure above correctly suggests that we can write a complex number using
trigonometric functions.
The trigonometric form of a complex number z  a  bi is
z  rcos  i sin    rcis , where r  a  bi  a 2  b 2 .
The r is called the modulus of z and an angle associated with z is called the argument
of z.
The trigonometric forms of a complex number can simplify the work of multiplying or
dividing complex numbers, as well as finding the powers or roots of complex numbers.
The theorem below helps us to multiply and divide complex numbers using their
trigonometric forms.
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Theorem for Products and Quotients of Complex Numbers
If trigonometric forms for two complex numbers z1 and z 2 are z1  r1 cos1  i sin 1 
and z2  r2 cos 2  i sin  2  then
1. z1  z2  r1  r2 cos1   2   i sin 1   2 
2.
z1 r1
 cos1   2   i sin 1   2  , for z 2  0
z 2 r2
Homework: 1-63 odd.
6
Section 4.4 DeMoivre’s Theorem and nth Roots of Complex Numbers
Recall that I mentioned that the trigonometric form of a complex number can simplify
finding the nth powers or nth roots of a complex number. The following theorems will
show you how to do this.
DeMoivre’s Theorem
n
For every integer n, r cos  i sin    r n cosn   i sin n 
Theorem on nth Roots
If z  r cos  i sin  is any nonzero complex number and if n is any positive integer,
then z has exactly n different nth roots w0 , w1 ,  , wn1 .
These roots, for  in radians are:
    2k 
   2k 
wk  n r cos
  i sin 

 n 
  n 
or equivalently, for  in degrees:
    360  k 
   360  k 
  i sin 

wk  n r cos
n
n



 
for k = 0, 1, … , n-1.
The nth roots of z all have absolute value n r and hence their geometric representations
lie on a circle of radius n r with center at the origin. Moreover, they are equally spaced
around the circle.
Special Applications
Now, consider the special case when z  1 . The n distinct nth roots of 1 are called the nth
roots of unity.
Finding the nth roots of a complex number z is equivalent to finding all the solutions of
the equation x n  z .
Homework: 1-29 odd.
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Section 4.5 Vectors
A scalar quantity is a quantity that has the property of size or magnitude but does not
possess the property of direction. Examples include: mass, length, time, density, energy,
area, volume, and temperature. The real number associated with a scalar quantity is
simply referred to as a scalar.
A vector quantity is a quantity that has both properties of magnitude and direction.
Examples include: velocity, wind movement, momentum, force, and displacement.
Vector quantities can be represented geometrically by a directed line segment (or arrow)
where the “tail” of the arrow is the initial point and the “tip” of the arrow is the terminal
point. These directed line segments are referred to as vectors.
The magnitude of a vector (also called the norm of a vector) is the length of the directed
line segment.
A vector of magnitude one is called a unit vector. There are two special unit vectors used
in the xy-plane. They are i  1, 0 and j  0,1 .
A vector of magnitude zero is called the zero vector. By definition, the zero vector is
0  0, 0 .
Vectors with the same magnitude and direction are said to be equivalent. Thus a vector
may be translated from one location to another, provided neither the magnitude nor the
direction is changed.
Notation of vectors
There are many ways to denote a vector. Four methods include:
 For a vector with initial point P and terminal point Q we can write PQ .
 Bold lowercase letters, such as a, b, u, v, or w, are used when the endpoints are
not specified. (In handwritten work we can write a, b, u, v, or w respectively).
 For a vector a with initial point at the origin and terminal point at a1 , a2  we can
denote the vector as a  a1 , a2 . The numbers a1 and a 2 are called the
horizontal and vertical components of the vector a, respectively.
 For a  a1 , a2 , it could be written using unit vectors as a  a1i  a2 j . This form
is called a linear combination of the unit vectors i and j.
Formula for the Magnitude of a Vector
The magnitude of the vector a  a1 , a2 , denoted by a , is given by a  a12  a22 .
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Arithmetic of Vectors
Addition, subtraction, and multiplication of vectors can be performed.
For vectors a  a1 , a2 , b  b1 , b2 , and scalar m,

a  b  a1  b1 , a2  b2

a  b  a1  b1 , a2  b2

ma  m  a1 , m  a2
Properties of Addition and Scalar Multiples of Vectors
For vectors a  a1 , a2 , b  b1 , b2 , and scalars m and n,
1.
2.
3.
4.
5.
6.
7.
8.
9.
ab  ba
a  b  c  a  b  c
a0 a
a   a  0
ma  b  ma  mb
m  na  ma  na
mna  mna  nma
1a  a
0a  0  m0
Formulas for Horizontal and Vertical Components of a Vector a
Let θ be an angle in standard position, measured from the positive x-axis to the vector
a  a1 , a2  a1i  a2 j . Then the horizontal and vertical components, a1 and a 2
respectively, can be found as follows.
a1  a cos 
a 2  a sin 
Resultant or Net Force
A force vector is a vector that describes the magnitude and direction of a force on an
object. When two or more forces are acting on an object, the resultant or net force is the
sum of the force vectors. When the net force vector is the zero vector, the object is said to
be in equilibrium.
Homework: 1-15 odd, 29-39 odd, and 49.
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Section 4.6 The Dot Product
When we multiply two vectors together we do not obtain another vector. Instead, we
obtain a scalar. This product is referred to as the dot product.
Definition of the Dot Product
Let a  a1 , a2  a1i  a2 j and b  b1 , b2  b1i  b2 j . The dot product of a and b,
denoted a b , is defined as follows.
a b  a1b1  a2b2
Properties of the Dot Product
If a, b, and c are vectors and m is a real number then:
2
1. a  a  a
2. a  b  b  a
3. a  b  c  a  b  a  c
4. ma b  ma  b  a  mb
5. 0 a  0
Definition of Parallel and Orthogonal Vectors
Let  be the angle between two nonzero vectors a and b. Then, by definition,
1. a and b are parallel if   0 or    .
2. a and b are orthogonal if  

.
2
Theorem on the Dot Product
If  is the angle between two nonzero vectors a and b, then a  b  a b cos .
Theorem on the Cosine of the Angle Between Vectors
If  is the angle between two nonzero vectors a and b, then cos  
ab
a b
Theorem on Orthogonal Vectors
Two vectors a and b are orthogonal if and only if a b  0 .
10
Definition of the Component of Vector a Along Vector b
Let  be the angle between two nonzero vectors a and b. The component of a along b
(also called the projection of a onto b), denoted by compb a (or projb a ) is defined as
follows.
compb a  a cos
Formula for compba
If a and b are nonzero vectors, then compb a 
ab
.
b
Work
One very important application of the dot product is the concept of work. If a constant
force F is applied to an object, moving it a distance d in the direction of the force, then,
by definition, the work W done is W  Fd . However, the direction of the force and the
direction of displacement are not always the same. In this case, the work W can be found
with the formula W  F d , where F is the force vector and d is the vector of
displacement.
The units of work differ depending on the measurement system you are working with.
The most common units of measurement are the foot-pound (ft-lb), dyne-centimeter
(erg), and newton-meter (joule).
Homework: 1-31 odd.
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