trigonometric system lesson of math on how to. solve triangle the unit cirlce is the guide to find the exact value of a triangle,it is the foundation on how to rely the exact value of pi ..finding the sin the cosine the tangent the secant the cosecant and the cotangent
5. Arc Length
Theorem. [Arc Length]
For a circle of radius r, a central angle of
µ radians subtends an arc whose length s
is
s = rµ
WARNING!
The angle must be given in radians
6. Arc Length
Example.
Problem: Find the length of the arc of a
circle of radius 5 centimeters subtended
by a central angle of 1.4 radians
Answer:
8. Radians vs. Degrees
Example. Convert each angle in
degrees to radians and each angle in
radians to degrees
(a) Problem: 45±
Answer:
(b) Problem: {270±
Answer:
(c) Problem: 2 radians
Answer:
10. Area of a Sector of a Circle
Theorem. [Area of a Sector]
The area A of the sector of a circle of
radius r formed by a central angle of µ
radians is
A = 1
2
r 2
µ
11. Area of a Sector of a Circle
Example.
Problem: Find the area of the sector of a
circle of radius 3 meters formed by an
angle of 45±. Round your answer to two
decimal places.
Answer:
WARNING!
The angle again must be given in
radians
12. Linear and Angular Speed
Object moving around a circle or
radius r at a constant speed
Linear speed: Distance traveled divided
by elapsed time
t = time
µ = central angle swept out in time t
s = rµ = arc length = distance traveled
v = s
t
13. Linear and Angular Speed
Object moving around a circle or
radius r at a constant speed
Angular speed: Angle swept out divided
by elapsed time
Linear and angular speeds are related
v = r!
! = µ
t
14. Linear and Angular Speed
Example. A neighborhood carnival
has a Ferris wheel whose radius is 50
feet. You measure the time it takes
for one revolution to be 90 seconds.
(a) Problem: What is the linear speed (in
feet per second) of this Ferris wheel?
Answer:
(b) Problem: What is the angular speed
(in radians per second)?
Answer:
15. Key Points
Basic Terminology
Measuring Angles
Degrees, Minutes and Seconds
Radians
Arc Length
Radians vs. Degrees
Area of a Sector of a Circle
Linear and Angular Speed
17. Unit Circle
Unit circle: Circle with radius 1
centered at the origin
Equation: x2 + y2 = 1
Circumference: 2¼
18. Unit Circle
Travel t units around circle, starting
from the point (1,0), ending at the
point P = (x, y)
The point P = (x, y) is used to define
the trigonometric functions of t
19. Trigonometric Functions
Let t be a real number and P = (x, y)
the point on the unit circle
corresponding to t:
Sine function: y-coordinate of P
sin t = y
Cosine function: x-coordinate of P
cos t = x
Tangent function: if x 0
20. Trigonometric Functions
Let t be a real number and P = (x, y)
the point on the unit circle
corresponding to t:
Cosecant function: if y 0
Secant function: if x 0
Cotangent function: if y 0
21. Exact Values Using Points on
the Circle
A point on the unit circle will satisfy
the equation x2 + y2 = 1
Use this information together with
the definitions of the trigonometric
functions.
22. Exact Values Using Points on
the Circle
Example. Let t be a real number and
P = the point on the unit
circle that corresponds to t.
Problem: Find the values of sin t, cos t,
tan t, csc t, sec t and cot t
Answer:
23. Trigonometric Functions of
Angles
Convert between arc length and
angles on unit circle
Use angle µ to define trigonometric
functions of the angle µ
24. Exact Values for Quadrantal
Angles
Quadrantal angles correspond to
integer multiples of 90± or of
radians
25. Exact Values for Quadrantal
Angles
Example. Find the values of the
trigonometric functions of µ
Problem: µ = 0 = 0±
Answer:
26. Exact Values for Quadrantal
Angles
Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 90±
Answer:
27. Exact Values for Quadrantal
Angles
Example. Find the values of the
trigonometric functions of µ
Problem: µ = ¼ = 180±
Answer:
28. Exact Values for Quadrantal
Angles
Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 270±
Answer:
35. Exact Values for Standard
Angles
Example. Find the values of the
following expressions
(a) Problem: sin(315±)
Answer:
(b) Problem: cos({120±)
Answer:
(c) Problem:
Answer:
36. Approximating Values Using a
Calculator
IMPORTANT!
Be sure that your calculator is in the
correct mode.
Use the basic trigonometric facts:
37. Approximating Values Using a
Calculator
Example. Use a calculator to find the
approximate values of the following.
Express your answers rounded to two
decimal places.
(a) Problem: sin 57±
Answer:
(b) Problem: cot {153±
Answer:
(c) Problem: sec 2
Answer:
38. Circles of Radius r
Theorem.
For an angle µ in standard position, let
P = (x, y) be the point on the terminal
side of µ that is also on the circle
x2 + y2 = r2. Then
39. Circles of Radius r
Example.
Problem: Find the exact values of each of
the trigonometric functions of an angle µ
if ({12, {5) is a point on its terminal
side.
Answer:
40. Key Points
Unit Circle
Trigonometric Functions
Exact Values Using Points on the
Circle
Trigonometric Functions of Angles
Exact Values for Quadrantal Angles
Exact Values for Standard Angles
Approximating Values Using a
Calculator
43. Domains of Trigonometric
Functions
Domain of sine and cosine functions is
the set of all real numbers
Domain of tangent and secant
functions is the set of all real
numbers, except odd integer multiples
of = 90±
Domain of cotangent and cosecant
functions is the set of all real
numbers, except integer multiples of
¼ = 180±
44. Ranges of Trigonometric
Functions
Sine and cosine have range [{1, 1]
{1 · sin µ · 1; jsin µj · 1
{1 · cos µ · 1; jcos µj · 1
Range of cosecant and secant is
({1, {1] [ [1, 1)
jcsc µj ¸ 1
jsec µj ¸ 1
Range of tangent and cotangent
functions is the set of all real numbers
45. Periods of Trigonometric
Functions
Periodic function: A function f with
a positive number p such that
whenever µ is in the domain of f, so is
µ + p, and
f(µ + p) = f(µ)
(Fundamental) period of f: smallest
such number p, if it exists
46. Periods of Trigonometric
Functions
Periodic Properties:
sin(µ + 2¼) = sin µ
cos(µ + 2¼) = cos µ
tan(µ + ¼) = tan µ
csc(µ + 2¼) = csc µ
sec(µ + 2¼) = sec µ
cot(µ + ¼) = cot µ
Sine, cosine, cosecant and secant have
period 2¼
Tangent and cotangent have period ¼
48. Signs of the Trigonometric
Functions
P = (x, y) corresponding to angle µ
Definitions of functions, where defined
Find the signs of the functions
Quadrant I: x > 0, y > 0
Quadrant II: x < 0, y > 0
Quadrant III: x < 0, y < 0
Quadrant IV: x > 0, y < 0
54. Pythagorean Identities
Example. Find the exact values of
each expression. Do not use a
calculator
(a) Problem: cos 20± sec 20±
Answer:
(b) Problem: tan2 25± { sec2 25±
Answer:
56. Even-Odd Properties
A function f is even if f({µ) = f(µ)
for all µ in the domain of f
A function f is odd if f({µ) = {f(µ)
for all µ in the domain of f
57. Even-Odd Properties
Theorem. [Even-Odd Properties]
sin({µ) = {sin(µ)
cos({µ) = cos(µ)
tan({µ) = {tan(µ)
csc({µ) = {csc(µ)
sec({µ) = sec(µ)
cot({µ) = {cot(µ)
Cosine and secant are even functions
The other functions are odd functions
58. Even-Odd Properties
Example. Find the exact values of
(a) Problem: sin({30±)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
62. Graphing Trigonometric
Functions
Graph in xy-plane
Write functions as
y = f(x) = sin x
y = f(x) = cos x
y = f(x) = tan x
y = f(x) = csc x
y = f(x) = sec x
y = f(x) = cot x
Variable x is an angle, measured in radians
Can be any real number
63. Graphing the Sine Function
Periodicity: Only need to graph on
interval [0, 2¼] (One cycle)
Plot points and graph
64. Properties of the Sine Function
Domain: All real numbers
Range: [{1, 1]
Odd function
Periodic, period 2¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Maximum value: y = 1, occurring at
Minimum value: y = {1, occurring at
66. Graphing the Cosine Function
Periodicity: Again, only need to graph
on interval [0, 2¼] (One cycle)
Plot points and graph
67. Properties of the Cosine
Function
Domain: All real numbers
Range: [{1, 1]
Even function
Periodic, period 2¼
x-intercepts:
y-intercept: 1
Maximum value: y = 1, occurring at
x = …, {2¼, 0, 2¼, 4¼, 6¼, …
Minimum value: y = {1, occurring at
x = …, {¼, ¼, 3¼, 5¼, …
69. Sinusoidal Graphs
Graphs of sine and cosine functions
appear to be translations of each
other
Graphs are called sinusoidal
Conjecture.
70. Amplitude and Period of
Sinusoidal Functions
Graphs of functions y = A sin x and
y = A cos x will always satisfy
inequality {jAj · y · jAj
Number jAj is the amplitude
71. Amplitude and Period of
Sinusoidal Functions
Graphs of functions y = A sin x and
y = A cos x will always satisfy
inequality {jAj · y · jAj
Number jAj is the amplitude
2
3
2
2 5
2
3
2
-4
-2
2
4
2
3
2
2 5
2
3
2
-4
-2
2
4
72. Amplitude and Period of
Sinusoidal Functions
Period of y = sin(!x) and
y = cos(!x) is
2
3
2
2 5
2
3
2
-4
-2
2
4
2
3
2
2 5
2
3
2
-4
-2
2
4
73. Amplitude and Period of
Sinusoidal Functions
Cycle: One period of y = sin(!x) or
y = cos(!x)
2
3
2
2 5
2
3
2
-4
-2
2
4
2
3
2
2 5
2
3
2
-4
-2
2
4
74. Amplitude and Period of
Sinusoidal Functions
Cycle: One period of y = sin(!x) or
y = cos(!x)
75. Amplitude and Period of
Sinusoidal Functions
Theorem. If ! > 0, the amplitude and
period of y = Asin(!x) and
y = Acos(! x) are given by
Amplitude = j Aj
Period = .
76. Amplitude and Period of
Sinusoidal Functions
Example.
Problem: Determine the amplitude and
period of y = {2cos(¼x)
Answer:
77. Graphing Sinusoidal Functions
One cycle contains four important
subintervals
For y = sin x and y = cos x these are
Gives five key points on graph
79. Finding Equations for
Sinusoidal Graphs
Example.
Problem: Find an equation for the graph.
Answer:
2
3
2
2 5
2
3
2
3
2
2
5
2
3
-6
-4
-2
2
4
6
80. Key Points
Graphing Trigonometric Functions
Graphing the Sine Function
Properties of the Sine Function
Transformations of the Graph of the
Sine Functions
Graphing the Cosine Function
Properties of the Cosine Function
Transformations of the Graph of the
Cosine Functions
81. Key Points (cont.)
Sinusoidal Graphs
Amplitude and Period of Sinusoidal
Functions
Graphing Sinusoidal Functions
Finding Equations for Sinusoidal
Graphs
84. Properties of the Tangent
Function
Domain: All real numbers, except odd
multiples of
Range: All real numbers
Odd function
Periodic, period ¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Asymptotes occur at
87. Graphing the Cosecant and
Secant Functions
Use reciprocal identities
Graph of y = csc x
88. Graphing the Cosecant and
Secant Functions
Use reciprocal identities
Graph of y = sec x
89. Key Points
Graphing the Tangent Function
Properties of the Tangent Function
Transformations of the Graph of the
Tangent Functions
Graphing the Cotangent Function
Graphing the Cosecant and Secant
Functions
91. Graphing Sinusoidal Functions
y = A sin(!x), ! > 0
Amplitude jAj
Period
y = A sin(!x { Á)
Phase shift
Phase shift indicates amount of shift
To right if Á > 0
To left if Á < 0
92. Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Determine amplitude jAj
Determine period
Determine starting point of one cycle:
Determine ending point of one cycle:
93. Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Divide interval into four
subintervals, each with length
Use endpoints of subintervals to find the
five key points
Fill in one cycle
94. Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Extend the graph in each direction to
make it complete
95. Graphing Sinusoidal Functions
Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
Answer:
(c) Problem: Find the phase shift
Answer:
96. Finding a Sinusoidal Function
from Data
Example. An experiment in a wind tunnel
generates cyclic waves. The following data is
collected for 52 seconds.
Let v represent the wind speed in feet per second
and let x represent the time in seconds.
Time (in seconds), x Wind speed (in feet per
second), v
0 21
12 42
26 67
41 40
52 20
97. Finding a Sinusoidal Function
from Data
Example. (cont.)
Problem: Write a sine equation that
represents the data
Answer:
98. Key Points
Graphing Sinusoidal Functions
Finding a Sinusoidal Function from
Data