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Unit Circle and
Radians
Radians
 Central angle: An angle whose vertex
is at the center of a circle
 Central angles subtend an arc on the
circle
Radians
 One radian is the measure of an
angle which subtends an arc with
length equal to the radius of the circle
Radians
IMPORTANT!
 Radians are dimensionless
 If an angle appears with no units, it
must be assumed to be in radians
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
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:
Radians vs. Degrees
 1 revolution = 2¼ radians = 360±
 180± = ¼ radians
 1± = radians
 1 radian =
¼
180
180
¼
±
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:
Radians vs. Degrees
 Measurements of common angles
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
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
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
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
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:
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
Trigonometric
Functions: Unit
Circle Approach
Section 5.2
Unit Circle
 Unit circle: Circle with radius 1
centered at the origin
 Equation: x2 + y2 = 1
 Circumference: 2¼
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
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
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
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.
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:
Trigonometric Functions of
Angles
 Convert between arc length and
angles on unit circle
 Use angle µ to define trigonometric
functions of the angle µ
Exact Values for Quadrantal
Angles
 Quadrantal angles correspond to
integer multiples of 90± or of
radians
Exact Values for Quadrantal
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = 0 = 0±
Answer:
Exact Values for Quadrantal
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 90±
Answer:
Exact Values for Quadrantal
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = ¼ = 180±
Answer:
Exact Values for Quadrantal
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 270±
Answer:
Exact Values for Quadrantal
Angles
Exact Values for Quadrantal
Angles
 Example. Find the exact values of
(a) Problem: sin({90±)
Answer:
(b) Problem: cos(5¼)
Answer:
Exact Values for Standard
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 45±
Answer:
Exact Values for Standard
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 60±
Answer:
Exact Values for Standard
Angles
 Example. Find the values of the
trigonometric functions of µ
Problem: µ = = 30±
Answer:
Exact Values for Standard
Angles
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:
Approximating Values Using a
Calculator
IMPORTANT!
 Be sure that your calculator is in the
correct mode.
 Use the basic trigonometric facts:
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:
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
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:
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
Key Points (cont.)
 Circles of Radius r
Properties of the
Trigonometric
Functions
Section 5.3
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±
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
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
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
Periods of Trigonometric
Functions
 Example. Find the exact values of
(a) Problem: sin(7¼)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
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
Signs of the Trigonometric
Functions
Signs of the Trigonometric
Functions
 Example:
Problem: If sin µ < 0 and cos µ > 0, name
the quadrant in which the angle µ lies
Answer:
Quotient Identities
 P = (x, y) corresponding to angle µ:
 Get quotient identities:
Quotient Identities
 Example.
Problem: Given and
, find the exact values of
the four remaining trigonometric
functions of µ using identities.
Answer:
Pythagorean Identities
 Unit circle: x2 + y2 = 1
 (sin µ)2 + (cos µ)2 = 1
sin2 µ + cos2 µ = 1
tan2 µ + 1 = sec2 µ
1 + cot2 µ = csc2 µ
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:
Pythagorean Identities
 Example.
Problem: Given that and that
µ is in Quadrant II, find cos µ.
Answer:
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
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
Even-Odd Properties
 Example. Find the exact values of
(a) Problem: sin({30±)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Fundamental Trigonometric
Identities
 Quotient Identities
 Reciprocal Identities
 Pythagorean Identities
 Even-Odd Identities
Key Points
 Domains of Trigonometric Functions
 Ranges of Trigonometric Functions
 Periods of Trigonometric Functions
 Signs of the Trigonometric Functions
 Quotient Identities
 Pythagorean Identities
 Even-Odd Properties
 Fundamental Trigonometric Identities
Graphs of the
Sine and Cosine
Functions
Section 5.4
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
Graphing the Sine Function
 Periodicity: Only need to graph on
interval [0, 2¼] (One cycle)
 Plot points and graph
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
2
3
2
2 5
2
3
2
-4
-2
2
4
Transformations of the Graph
of the Sine Functions
 Example.
Problem: Use the graph of y = sin x to
graph
Answer:
Graphing the Cosine Function
 Periodicity: Again, only need to graph
on interval [0, 2¼] (One cycle)
 Plot points and graph
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, …
2
3
2
2 5
2
3
2
-4
-2
2
4
 Example.
Problem: Use the graph of y = cos x to
graph
Answer:
Transformations of the Graph
of the Cosine Functions
Sinusoidal Graphs
 Graphs of sine and cosine functions
appear to be translations of each
other
 Graphs are called sinusoidal
 Conjecture.
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
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
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
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
Amplitude and Period of
Sinusoidal Functions
 Cycle: One period of y = sin(!x) or
y = cos(!x)
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 = .
Amplitude and Period of
Sinusoidal Functions
 Example.
Problem: Determine the amplitude and
period of y = {2cos(¼x)
Answer:
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
2
3
2
2 5
2
3
2
-4
-2
2
4
Graphing Sinusoidal Functions
 Example.
Problem: Graph y = {3cos(2x)
Answer:
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
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
Key Points (cont.)
 Sinusoidal Graphs
 Amplitude and Period of Sinusoidal
Functions
 Graphing Sinusoidal Functions
 Finding Equations for Sinusoidal
Graphs
Graphs of the
Tangent, Cotangent,
Cosecant and Secant
Functions
Section 5.5
Graphing the Tangent
Function
 Periodicity: Only need to graph on
interval [0, ¼]
 Plot points and graph
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
2
3
2
2 5
2
3
2
-8
-6
-4
-2
2
4
6
8
Transformations of the Graph
of the Tangent Functions
 Example.
Problem: Use the graph of y = tan x to
graph
Answer:
Graphing the Cotangent
Function
 Periodicity: Only need to graph on
interval [0, ¼]
Graphing the Cosecant and
Secant Functions
 Use reciprocal identities
 Graph of y = csc x
Graphing the Cosecant and
Secant Functions
 Use reciprocal identities
 Graph of y = sec x
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
Phase Shifts;
Sinusoidal Curve
Fitting
Section 5.6
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
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:
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
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
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:
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
Finding a Sinusoidal Function
from Data
 Example. (cont.)
Problem: Write a sine equation that
represents the data
Answer:
Key Points
 Graphing Sinusoidal Functions
 Finding a Sinusoidal Function from
Data

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unit_circle_lesson_in trigonometric functions

  • 2. Radians  Central angle: An angle whose vertex is at the center of a circle  Central angles subtend an arc on the circle
  • 3. Radians  One radian is the measure of an angle which subtends an arc with length equal to the radius of the circle
  • 4. Radians IMPORTANT!  Radians are dimensionless  If an angle appears with no units, it must be assumed to be in radians
  • 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:
  • 7. Radians vs. Degrees  1 revolution = 2¼ radians = 360±  180± = ¼ radians  1± = radians  1 radian = ¼ 180 180 ¼ ±
  • 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:
  • 9. Radians vs. Degrees  Measurements of common angles
  • 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:
  • 29. Exact Values for Quadrantal Angles
  • 30. Exact Values for Quadrantal Angles  Example. Find the exact values of (a) Problem: sin({90±) Answer: (b) Problem: cos(5¼) Answer:
  • 31. Exact Values for Standard Angles  Example. Find the values of the trigonometric functions of µ Problem: µ = = 45± Answer:
  • 32. Exact Values for Standard Angles  Example. Find the values of the trigonometric functions of µ Problem: µ = = 60± Answer:
  • 33. Exact Values for Standard Angles  Example. Find the values of the trigonometric functions of µ Problem: µ = = 30± Answer:
  • 34. Exact Values for Standard Angles
  • 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
  • 41. Key Points (cont.)  Circles of Radius r
  • 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
  • 47. Periods of Trigonometric Functions  Example. Find the exact values of (a) Problem: sin(7¼) Answer: (b) Problem: Answer: (c) Problem: Answer:
  • 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
  • 49. Signs of the Trigonometric Functions
  • 50. Signs of the Trigonometric Functions  Example: Problem: If sin µ < 0 and cos µ > 0, name the quadrant in which the angle µ lies Answer:
  • 51. Quotient Identities  P = (x, y) corresponding to angle µ:  Get quotient identities:
  • 52. Quotient Identities  Example. Problem: Given and , find the exact values of the four remaining trigonometric functions of µ using identities. Answer:
  • 53. Pythagorean Identities  Unit circle: x2 + y2 = 1  (sin µ)2 + (cos µ)2 = 1 sin2 µ + cos2 µ = 1 tan2 µ + 1 = sec2 µ 1 + cot2 µ = csc2 µ
  • 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:
  • 55. Pythagorean Identities  Example. Problem: Given that and that µ is in Quadrant II, find cos µ. 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:
  • 59. Fundamental Trigonometric Identities  Quotient Identities  Reciprocal Identities  Pythagorean Identities  Even-Odd Identities
  • 60. Key Points  Domains of Trigonometric Functions  Ranges of Trigonometric Functions  Periods of Trigonometric Functions  Signs of the Trigonometric Functions  Quotient Identities  Pythagorean Identities  Even-Odd Properties  Fundamental Trigonometric Identities
  • 61. Graphs of the Sine and Cosine Functions Section 5.4
  • 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
  • 65. 2 3 2 2 5 2 3 2 -4 -2 2 4 Transformations of the Graph of the Sine Functions  Example. Problem: Use the graph of y = sin x to graph Answer:
  • 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, …
  • 68. 2 3 2 2 5 2 3 2 -4 -2 2 4  Example. Problem: Use the graph of y = cos x to graph Answer: Transformations of the Graph of the Cosine Functions
  • 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
  • 78. 2 3 2 2 5 2 3 2 -4 -2 2 4 Graphing Sinusoidal Functions  Example. Problem: Graph y = {3cos(2x) Answer:
  • 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
  • 82. Graphs of the Tangent, Cotangent, Cosecant and Secant Functions Section 5.5
  • 83. Graphing the Tangent Function  Periodicity: Only need to graph on interval [0, ¼]  Plot points and graph
  • 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
  • 85. 2 3 2 2 5 2 3 2 -8 -6 -4 -2 2 4 6 8 Transformations of the Graph of the Tangent Functions  Example. Problem: Use the graph of y = tan x to graph Answer:
  • 86. Graphing the Cotangent Function  Periodicity: Only need to graph on interval [0, ¼]
  • 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