C4

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

C

γ

b a γ a

h

h b

α β α β

A B B

c A c



A is acute 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. Two angles and any side (AAS or ASA).

2. Two sides and an angle opposite one of them (SSA).

3. Three sides (SSS).

4. 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

sin   sin   sin  

If ABC is a triangle with sides a, b, and c, with height h, then   .

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 Necessary Condition Number of Solutions

ah None

ah One

α is Acute

ab One

hab Two

ab None

α is Obtuse

ab 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



is one-half the perimeter; that is s  a  b  c  .

1

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

Real axis

a





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.







5

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 z 2  r2 cos 2  i sin  2  then



1. z1  z 2  r1  r2 cos1   2   i sin 1   2 





 cos1   2   i sin 1   2  , for z 2  0

z1 r1

2.

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

For every integer n, r cos  i sin    r n cosn   i sin n 

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 

wk  n r cos

   i sin 

  



  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.







7

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 , a 2  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  a 2 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  a2 .

2









8

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. ab  ba

2. a  b  c  a  b  c

3. a0 a

4. a   a   0

5. ma  b   ma  mb

6. m  na  ma  na

7. mn a  mna  nma

8. 1a  a

9. 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.









9

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  a2 b2





Properties of the Dot Product

If a, b, and c are vectors and m is a real number then:

1. a  a  a

2





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

a b

If  is the angle between two nonzero vectors a and b, then cos 

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 comp b a (or projb a ) is defined as

follows.



compba  a cos





Formula for compba

a b

If a and b are nonzero vectors, then comp b a  .

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.









11