Section 2.1 Angles
Trigonometry is the branch of mathematics that deals with the relationships between
angles and sides of triangles as well as the theory of the periodic functions connected
with them. The name is derived from two Greek words: trigonon (meaning triangle) and
metria (meaning measurement).
There are many definitions we need to begin our discussion of trigonometry.
A line is determined by two points, say A and B, and goes off to infinity in two
directions.
A ray is “half a line.” It has an end point and goes off to infinity in only one direction.
A line segment is a portion of a line and has two endpoints.
An angle is created when either:
1. the endpoints of two rays meet each other, or
2. endpoints of two line segments meet each other.
The point where the two rays or line segments meet is called the vertex of the angle.
In trigonometry, we often think of angles as rotations of rays or line segments. The
starting position of the ray is called the initial side of the angle and the final position of
the ray after the rotation is called the terminal side of the angle. The point of rotation is
the vertex of the angle. Many angles have the same initial and terminal sides. Such angles
are called coterminal.
If we place an angle in the Cartesian coordinate system, with the initial side of the angle
coinciding with the positive x-axis and the vertex at the origin, we say that the angle is in
standard position. Whichever quadrant the terminal side of the angle resides, we say
that the angle is in that quadrant. If the terminal side of the angle resides on an axis, the
angle is said to be quadrantal.
The measurement of an angle is determined by two factors.
1. The direction of rotation.
a. Counterclockwise rotation produces a positive angle.
b. Clockwise rotation produces a negative angle.
2. The amount of rotation which is measured in either degrees or radians
a. One complete counterclockwise revolution is defined to be equal to 360°.
b. Alternatively, one complete counterclockwise revolution is defined to be
equal to 2  6.28 radians.
We can use these facts to find coterminal angles. Just add or subtract whole number
multiples of 360° or 2 radians to the angle.
1
We can also use these facts to convert an angle measured in degrees to radians and vice
versa. Since 360   2 radians, we can use this as a conversion rate or simplify it first to
the rate of 180    radians.
Degrees, Minutes, and Seconds
Degrees can be expressed as decimals or the decimal part can be further broken down
into minutes (denoted by ') and seconds (denoted by "). There are 60 minutes in one
degree and 60 seconds in one minute.
Types of Angles
A right angle is an angle whose measure is exactly 90°.
An acute angle is an angle that measures between 0° and 90°.
An obtuse angle is an angle that measures between 90° and 180°.
A straight angle is an angle whose measure is exactly 180°.
Complementary angles are two angles whose sum is exactly 90°.
Supplementary angles are two angles whose sum is exactly 180°.
Circles
A circle is the set of all points in the plane that are a fixed distance from a given point,
called the center, O. The distance of a line segment connecting the center of the circle to
a point on the circle is called the radius, r, of the circle. The length of a line segment
joining two points on the circle and the center of the circle is called the diameter, d, of
the circle. Note: d  2r .
A central angle,  , of a circle is an angle whose vertex is at the center of the circle. For
any central angle  we say that the arc of the circle subtends (or intersects)  . The
length of an arc (i.e. arc length), s, is found by the formula s  r , where r is the radius
of the circle and  is the measure of the central angle in radians.
The circumference, C, of a circle can be found with the formula C  2r  d .
The area of a circle, A, is found by the formula A  r 2 . If we only need to know the
1
area of a circular sector of the circle we can use the formula A  r 2 , where r is the
2
radius of the circle and  is the measure of the central angle in radians.
The angular speed of a wheel that is rotating at a constant rate is the angle generated in
one unit of time by a line segment from the center of the wheel to a point P on the
circumference. If the wheel is moving at a rate R in revolutions per minute (rpms) then
the angular speed  , in radians per minute, can be found by the formula   2R .
2
The linear speed (or velocity), v, of a point P on the circumference of a wheel is the
distance that P travels per unit time. Linear speed can be found by the formula v  r ,
where r is the radius of the wheel and  is the angular velocity of the wheel.
Special Angles
There are some special angles that you will need to learn their corresponding radian and
degree measurements. They are listed for you in the table on page 96 of your textbook.
Start now to memorize them.
Homework: 1-39, odd, 45, 47
3
Section 2.2 The Trigonometric Functions
Consider the equation x 2  y 2  1 . When graphed, this equation produces a circle of
radius one centered at the origin. This circle is called the unit circle.
Imagine a real number line is “wrapped” around the circle, with positive numbers
corresponding to counterclockwise wrapping and negative numbers corresponding to
clockwise wrapping. The value of 0 on the number line is located at the point (1, 0) on
the circle.
On this number line, each real number t will correspond to an x and y value (i.e. an
ordered pair (x, y)) on the circle that are used to define the six trigonometric functions
of t, which are: sine, cosine, tangent, cotangent, secant, and cosecant. They are
abbreviated as follows: sin, cos, tan, cot, sec, and csc respectively.
The six trigonometric functions of t are defined as follows:
sin t  y
csc t 
1
y
cos t  x
sec t 
1
x
y
x
x
cot t 
y
tan t 
Because these functions are defined on the unit circle, they are also called circular
functions.
Domain and Range of the Trigonometric Functions
The domain of sine and cosine is the set of all real numbers.
The domain of tangent and secant is all real numbers except 
 2  n for any integer n.
The domain of cotangent and cosecant is all real numbers except n for any integer n.
The range of sine and cosine is [ – 1, 1].
The range of tangent and cotangent is all real numbers.
The range of secant and cosecant is (,  1]  [1, ) .
Identities
An identity is an algebraic equation that states that two expressions are equal to each
other for all values of the variables in the expressions. In trigonometry there are many
identities. The fundamental identities in trigonometry consist of the reciprocal
identities, tangent and cotangent identities, and Pythagorean identities. They follow
below.
4
Reciprocal Identities
1
1
sin t 
cos t 
csc t
sec t
1
1
csc t 
sec t 
sin t
cos t
1
cot t
1
cot t 
tan t
tan t 
Tangent and Cotangent Identities
sin t
cos t
tan t 
cot t 
cos t
sin t
The Pythagorean Identities
sin 2 t  cos 2 t  1
1  tan 2 t  sec 2 t
1  cot 2 t  csc 2 t
Homework: 1-71 odd
5
Section 2.3 Graphs of the Trigonometric Functions
Consider a point, P (x, y) on the unit circle. Since cos t  x and sin t  y we could write
the point as Pcos t , sin t  . Once we make one complete revolution, the values of the
ordered pairs repeat. As a result, sin t  2n  sin t and cost  2n  cos t for any
integer n.
When functions repeat values over fixed intervals, the function is said to be periodic. The
formal definition of a periodic function follows.
A function f is said to be periodic (or cyclical) if there exists a positive real number k
such that f t  k   f t  for every t in the domain of f. The least such positive real
number k, if it exists, is the period of f.
Graphing Trigonometric Functions
When we graph a trigonometric function, we graph the value of the function versus t. For
example, if we were to graph the function y  sin t , we would need to create a table of
values with ordered pairs t , sin t  and graph them in a ty-plane.
The sine and cosine functions have a period of 2 (i.e. the pattern repeats every 2
radians). Thus, the interval 0  t  2 is referred to as one cycle and the pattern produced
is called the sine wave or cosine wave, respectively.
The cosecant and secant functions also have period of 2 . However, the tangent and
cotangent functions have period  . We can use these facts to aid in graphing them.

 n for
2
any integer n; and the cotangent and cosecant functions have vertical asymptotes at
t  n for any integer n.
In addition, the tangent and secant functions have vertical asymptotes at t 
Since t can be negative, we can also graph a trigonometric function over negative values
of t. However, due to the periodic nature of the trigonometric functions we can use some
rules to simplify the work in creating a table of values of negative t. These formulas are
below.
Formulas for Negatives
sin  t    sin t
cos t   cos t
csc t    csc t
sec t   sec t
tan  t    tan t
cot  t    cot t
Even and Odd Functions
A function is said to be even if it is symmetric with respect to the y-axis.
6
A function is said to be odd if it is symmetric with respect to the origin.
The cosine and secant functions are even.
The sine, tangent, cotangent, and cosecant functions are odd.
A table summarizing features of the six trigonometric functions we have studied so far is
on page 124 of your textbook. You should refer to it because it will help you understand
the features of these graphs.
Limits
A limit of a function is the value the dependent variable approaches near a value of the
dependent variable. There are right-hand limits and left-hand limits. For example, the

, sin t  1 ” means in English “As the independent
mathematical statement “As t 
2

variable t approaches
from the right, the function sin t approaches one.” The value of
2
one in the preceding statement would be considered a right-hand limit, since we

approached
from the right.
2
We can use graphs of the trigonometric functions to find their limits for given values of
the independent variable.
Homework: 1-35, odd.
7
Section 2.4 Trigonometric Functions of Angles
Previously we defined the trigonometric functions based on the unit circle, which has a
radius of one. These definitions are convenient when we can use them because the angle
 is equal to t on the unit circle. In other words, on the unit circle sin t  sin  .
However, we still need to be able to find the values of the trigonometric functions when
we are working with circles whose radii are not equal to one. In these situations, it is best
to refer to the independent variable by the central angle  . The following definitions of
the trigonometric functions must be used when the radius is not equal to one (i.e. r  1 ).
Definition
Let  be an angle in standard position on a rectangular coordinate system, and let
Qx, y  be any point other than the origin O on the terminal side of  .
If d O, Q   r  x 2  y 2 , then
y
r
r
csc  
y
sin  
x
r
r
sec  
x
cos  
y
x
x
cot  
y
tan  
Trigonometric Functions of Acute Angles
When the central angle  is acute (between 0° and 90°, or between 0 and 
radians)
2
we can define the trigonometric functions in terms of the sides of the right triangle that
can be formed by drawing a vertical line to the x-axis from the end of the terminal side of
the angle. These definitions follow.
opp
hyp
hyp
csc  
opp
sin  
adj
hyp
hyp
sec  
adj
cos  
opp
adj
adj
cot  
opp
tan  
There are times when these definitions are very useful.
Homework: 1-37 odd.
8
Section 2.5 Values of the Trigonometric Functions
Definition
Let  be a nonquadrantal angle in standard position. The reference angle for  is the
acute angle  R that the terminal side of  makes with the x-axis.
To determine the reference angle for  , we must consider which quadrant the terminal
side of  resides.




Quadrant I:  R   .
Quadrant II:  R  180       
Quadrant III:  R    180     
Quadrant IV:  R  360    2  
We can use reference angles to find the exact values of the trigonometric functions by
using the following theorem.
Theorem
If  is a nonquadrantal angle in standard position, then to find the value of a
trigonometric function at  , find is value for the reference angle  R and prefix the
appropriate sign (positive or negative).
When only an approximate value for the trigonometric function is needed you may use
your calculator to find it.
Solving Trigonometric Equations for θ
To solve a trigonometric equation for  will involve the use of the inverse trigonometric
functions: sin 1 , cos 1 , and tan 1 , our calculators, and understanding of the periodic
nature of these functions.
Homework: 1-39 odd.
9
Section 2.6 Trigonometric Graphs
The graphs of the trigonometric functions y  sin x and y  cos x can be vertically or
horizontally compressed or stretched, or they may be vertically or horizontally shifted
with simple changes to the basic functions above. We consider each case below.
Vertical Compression or Stretching
Consider the transformations y  a sin x and y  a cos x .
 If a  1 the graphs are vertically stretched.

If 0  a  1 the graphs are vertically compressed.
In either case, a is referred to the amplitude of the graph or function and represents
the maximum height of the graph from a given reference level (usually y  0 ).
Horizontal Compression or Stretching
Consider the additional transformations y  a sin bx and y  a cosbx .
 If b  1 the graphs are horizontally compressed.

If 0  b  1 the graphs are horizontally stretched.
In either case, b changes the period of these functions from 2 to a new period of
2
.
b
Horizontal Shifts
Consider the additional transformations y  a sin bx  c and y  a cosbx  c .
 If c  0 the graph is shifted horizontally to the left.
 If c  0 the graph is shifted horizontally to the right.
c
. A shift
b
to the left produces a negative phase shift and a shift to the right produces a positive
phase shift.
The amount of the shift is called the phase shift and is found with the ratio
When a transformation includes a change in period and/or a phase shift, we can still find
an interval that contains exactly one cycle of the sine or cosine wave by solving the
inequality 0  bx  c  2 . If we do this, it will greatly aid in the graphing of the
transformed function.
Vertical Shifts
Consider the final transformations y  a sin bx  c  d and y  a cosbx  c  d .
 If d  0 the graph is shifted vertically upward by d units.
 If d  0 the graph is shifted vertically downward by d units.
10
Finding an Equation of a Sine Wave From Its Graph
To find the equation of a sine wave of the form y  a sin bx  c from its graph, where
the restrictions of a  0, b  0, and c  0 are given:
1. Find a, the amplitude of the graph by finding the maximum y-value on the graph.
2. Find b, by determining the period of the wave and then solving the equation
2
period 
.
b
3. Find c, by noting how far the graph has been moved to the left (which is the phase
c
shift) then solving the equation phase shift 
. Note: Since the restriction was
b
given that c  0 , the phase shift will be negative.
Homework: 1-43 odd.
11
Section 2.7 Additional Trigonometric Graphs
The other four trigonometric functions – tangent, cotangent, secant and cosecant – can
also be transformed. Since these functions do not have maximum or minimum values
they do not possess the feature of amplitude. However, we can alter their periods and
create phase shifts. A theorem on transforming the tangent function follows.
Theorem
If y  a tan bx  c for nonzero numbers a and b, then
1. the period is
c

and the phase shift is 
b
b
2. successive vertical asymptotes for the graph may be found by solving the


inequality   bx  c  .
2
2
There are many other alterations one can make with a trigonometric function. For
example, we could add, subtract, multiply, or divide them. In addition, we could add,
subtract, multiply, or divide them by a non-trigonometric function or take their absolute
value – just to name a few options.
Homework: 1, 9, 11, 13, 15, 17, 49, 55, 57, 59, 73.
12
Section 2.8 Applied Problems
Solving a Right Triangle
To solve a right triangle means to find the lengths of all of its sides and the measurement
of all of its angles. If we know the lengths of two of its sides or if we know the length of
one side and the measure of one of its acute angles we can solve any right triangle by
using the Pythagorean Theorem and the definitions of the trigonometric functions with
respect to a right triangle.
Notation
The following drawing represents how we will label the parts of ABC , with   90  .
B
β
c
A
a
γ
α
b
C
Angles of Elevation/Depression
If an observer at point X sights an object, then the angle that the line of sight makes with
the horizontal line l from point X is the angle of elevation of the object, if the object is
above the horizontal line, or the angle of depression of the object, if the object is below
the horizontal line.
Navigation or Surveying Applications
The direction or bearing from a point P to a point Q is specified by stating the acute
angle that segment PQ makes with the north-south line through P. We also state whether
Q is north or south and east or west of P. The notation always lists N or S first, to indicate
north or south, then the acute angle, and finally W or E, to indicate west or east.
For example: N 30° E, which is read “thirty degrees east of north.”
13
Harmonic Motion
The periodic nature of the trigonometric functions is useful for describing the motion of a
point on an object that vibrates, oscillates, rotates, or is moved by wave motion.
Assuming the ideal conditions of perfect elasticity and no friction or air resistance of such
an object we way that the object possesses simple harmonic motion.
Definition
A point that moves on a coordinate line is in simple harmonic motion if its distance d
from the origin at time t is given by either
d  a sin  t  or d  a cos t 
where a and  are real numbers such that   0 .
The motion has amplitude a , period
2

, and frequency

.
2
Homework: 1-15 odd, 25-31 odd, 61, 67-71 odd.
14