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Trigonometry
Cloud County Community College
Spring, 2012
Instructor: Timothy L. Warkentin
Chapter 2: Trigonometric Functions
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Angles and Arcs
Right Triangle Trigonometry
Trigonometric Functions of Any Angle
Trigonometric Functions of Real Numbers
Graphs of the Sine and Cosine Functions
Graphs of the other Trigonometric Functions
Graphing Techniques
Harmonic Motion – An Application of the Sine and
Cosine Functions
Chapter 2 Overview
• Chapter 2 begins by dealing with the fundamentals of angular
measurement and the relationships between angular and
linear quantities. Trigonometric values are then defined first as
ratios of sides in right triangles, then as ratios of coordinates
on the terminal side of any angle and finally as coordinates of
points on the unit circle. Even though a precise definition is
given for each trigonometric function no method of generally
calculating their values is given (until Calculus 3). Thus
trigonometry becomes the study of the properties of the
functions themselves and evaluation is left to computing
machines. The transformation and graphing of trigonometric
functions is covered in the last half of chapter 2.
2.1: Angles and Arcs 1
•
•
•
•
•
•
•
Degree Measure
Complementary and Supplementary Angles. Example 1
Coterminal Angles. Example 2
Radian: The angle subtended by one radius of arc.
Degrees ↔ Radian Conversions. Examples 3 & 4
Revolutions ↔ Radian Conversions. Example 7
Linear and Angular Relationships Examples 5, 6, & 8
Distance
Velocity
Acceleration
Linear
x
v
a
Link
x=θr
v=ωr
a=αr
Angular
θ
ω
α
2.2: Right Angle Trigonometry 1
• Right Triangle Trigonometric definitions: Examples 1, 2,
4, 5 & 6
opp
hyp
sin[  ] 
, csc[ ] 
hyp
opp
adj
hyp
cos[ ] 
, sec[ ] 
hyp
adj
opp
adj
tan[ ] 
, cot[ ] 
adj
opp
2.2: Right Angle Trigonometry 2
• Reference Triangles Example 3
2.3: Trigonometric Functions of Any Angle 1
• Any Angle Trigonometric definitions: Example 1
y
r
sin[  ]  , csc[ ] 
r
y
x
cos[ ]  , sec[ ] 
r
y
tan[ ]  , cot[ ] 
x
r
x
x
y
where r  x 2  y 2
• Quadrantal angles and Quadrant signs. Example 2
• Reference angles. Examples 3 & 4
2.4: Trigonometric Functions of Real Numbers 1
• The Wrapping function maps Arc Lengths = Angles (real
numbers) to points (x=cos[t], y=sin[t]) on the Unit Circle.
Example 1
• Real Number Trigonometric definitions: Examples 2 & 3
1
sin[  ]  y, csc[ ] 
y
1
cos[ ]  x, sec[ ] 
x
y
x
tan[ ]  , cot[ ] 
x
y
where x and y are Coordinate s of Points on the Unit Circle.
2.4: Trigonometric Functions of Real Numbers 2
•
•
•
•
The Domain and Range of the Trigonometric Functions.
Even and Odd Trigonometric Functions. Example 4
Periods of the Trigonometric Functions.
Trigonometric Identities. Examples 5-7
1
1
1
sin[ t ] 
, cos[t ] 
, tan[ t ] 
csc[t ]
sec[t ]
cot[t ]
sin[ t ]
cos[t ]
tan[ t ] 
, cot[t ] 
cos[t ]
sin[ t ]
sin 2 t  cos 2 t  1, 1  cot 2 t  csc 2 t , tan 2 t  1  sec 2 t
2.5: Graphs of the Sine and Cosine Functions 1
• Transforming a function: Class Example
1
y  a f [ ( x  h)]  k
w
–
–
–
–
a is the vertical scaling factor.
w is the horizontal scaling factor.
h is the horizontal shift.
k is the vertical shift.
2.5: Graphs of the Sine and Cosine Functions 2
• Sine and Cosine Curves: Periods, Quadrants,
Quadrantal Angles and Reference Angles.
2.5: Graphs of the Sine and Cosine Functions 3
• Transforming a periodic function:
y  A f [ ( x  h)]  k
–
–
–
–
A is the amplitude.
ω is the angular frequency.
h is the phase shift.
k is the vertical offset.
• The period of the function is its normal period divided by
the angular frequency.
period 
normal period

2.5: Graphs of the Sine and Cosine Functions 4
• Graphing a Trigonometric Function. Examples 1-7
1.
2.
3.
4.
5.
6.
7.
Write the equation in Standard Form and identify A, ω, h and k.
Compute the period and quarter period.
Draw the envelope (if possible) and center lines.
Find a divisor compatible (worst case is 1) with both the quarter
period and the horizontal offset. Mark the x-axis and center line in
these units.
Mark the horizontal offset point and quarter periods from that point
in both directions on the center line. Draw in any vertical
asymptotes.
For each quarter period point mark the appropriate height of
function on the graph. When graphing the cotangent, secant and
cosecant functions use the reciprocal of values from the tangent,
cosine and sine functions.
Connect the marks.
2.6: Graphs of the Other Trigonometric Functions 1
• Tangent and Cotangent Curves: Periods, Quadrants,
Quadrantal Angles and Reference Angles.
2.6: Graphs of the Other Trigonometric Functions 2
• Secant and Cosecant Curves: Periods, Quadrants,
Quadrantal Angles and Reference Angles.
2.6: Graphs of the Other Trigonometric Functions 3
• Graphing a Trigonometric Function. Examples 1-5
Write the equation in Standard Form and identify A, ω, h and k.
Compute the period and quarter period.
Draw the envelope (if possible) and center lines.
Find a divisor compatible (worst case is 1) with both the quarter
period and the horizontal offset. Mark the x-axis and center line
in these units.
5. Mark the horizontal offset point and quarter periods from that
point in both directions on the center line. Draw in any vertical
asymptotes.
6. For each quarter period point mark the appropriate height of
function on the graph. When graphing the cotangent, secant
and cosecant functions use the reciprocal of values from the
tangent, cosine and sine functions.
7. Connect the marks.
1.
2.
3.
4.
2.7: Graphing Techniques 1
• Graphing a Trigonometric Function. Examples 1-6
1.
2.
3.
4.
5.
6.
7.
Write the equation in Standard Form and identify A, ω, h and k.
Compute the period and quarter period.
Draw the envelope (if possible) and center lines.
Find a divisor compatible (worst case is 1) with both the quarter
period and the horizontal offset. Mark the x-axis and center line in
these units.
Mark the horizontal offset point and quarter periods from that point
in both directions on the center line. Draw in any vertical
asymptotes.
For each quarter period point mark the appropriate height of
function on the graph. When graphing the cotangent, secant and
cosecant functions use the reciprocal of values from the tangent,
cosine and sine functions.
Connect the marks.
2.7: Graphing Techniques 2
• Addition of Ordinates (calculator). Examples 7 & 8
• Damping Factors (calculator). Example 9
2.8: Harmonic Motion 1
• If the sinusoidal curve (sine or cosine) is shifted so that it
begins at the origin and is centered on the x-axis then
y  A sin[  (t  h)]  k or y  A cos[ (t  h)]  k
become
y  A sin[ t ] or y  A cos[t ].
• The frequency of a sinusoidal function is defined to be
the reciprocal of the period. The phase shift is then
1
2
f   
 2f .
p
p
and the sinusoidal equations reduce to
Example 1
y  A sin[ 2f t ] or y  A cos[ 2f t ].
2.8: Harmonic Motion 2
• Simple Harmonic Motion: periodic motion induced by a
restoring force that is directly proportional to the
displacement from the rest position. For springs this is
mathematically represented by Hooke’s Law: F = - k x.
• Physics shows that the frequency of oscillation for a
mass-spring system is
Example 2
1
f 
2
k
.
m
• Damped Harmonic Motion. Example 3
• Mass-Spring Experiment.