P.5 Angles and Arcs

1
P.5 Angles and Arcs

• Basic Terminology
• Two distinct points A and B determine the line
  AB.
• The portion of the line including the points A
  and B is the line segment AB.
• The portion of the line that starts at A and
  continues through B is called ray AB.
• An angle is formed by rotating a ray, the initial
  side, around its endpoint, the vertex, to a
  terminal side.

2
P.5 Degree Measure

• Degree Measure
• Developed by the Babylonians around 4000 yrs ago.
• Divided the circumference of the circle into 360
  parts. One possible reason for this is because
  there are approximately that number of days in a
  year.
• There are 360 in one rotation.
• An acute angle is an angle between 0 and 90.
• A right angle is an angle that is exactly 90.
• An obtuse angle is an angle that is greater than
  90 but less than 180.
• A straight angle is an angle that is exactly 180.

3
P.5 Finding Measures of Complementary and
Supplementary Angles

• If the sum of two positive angles is 90, the
  angles are called complementary.
• If the sum of two positive angles is 180, the
  angles are called supplementary.
• Example Find the measure of each angle in the
  given figure.
• (a) (b)

(Supplementary angles)
(Complementary angles)
Angles are 60 and 30 degrees
Angles are 72 and 108 degrees
4
P.5 Coterminal Angles

• Quadrantal Angles are
• angles in standard
• position (vertex at the
• origin and initial side
• along the positive x-
• axis) with terminal sides
• along the x or y axis,
• i.e. 90, 180, 270, etc.
• Coterminal Angles are
• angles that have the same
• initial side and the same
• terminal side.

5
P.5 Finding Measures of Coterminal Angles

• Example Find the angles of smallest possible
  positive
• measure coterminal with each angle.
• (a) 908 (b) 75
• Solution Add or subtract 360 as many times as
• needed to get an angle between 0 and 360.
• (a)
• (b)
• Let n be an integer, we have an infinite number
  of
• coterminal angles e.g. 60 n 360.

6
P.5 Radian Measure

• The radian is a real number, where the degree is
  a unit of measurement.
• The circumference of a circle, given by C 2? r,
  where r is the radius of the circle, shows that
  an angle of 360º has measure 2? radians.

An angle with its vertex at the center of a
circle that intercepts an arc on the circle
equal in length to the radius of the circle has
a measure of 1 radian.
7
P.5 Converting Between Degrees and Radians

• Multiply a radian measure by 180º/? and simplify
  to convert to degrees. For example,
• Multiply a degree measure by ? /180º and simplify
  to convert to radians. For example,

8
P.5 Converting Between Degrees and Radians With
the Graphing Calculator

• Example Convert 249.8º to radians. (under 2nd,
  APPS)
• Solution
• Put the calculator in radian mode.
• Example Convert 4.25 radians to degrees.
• Solution
• Put the calculator in degree mode.

9
P.5 Equivalent Angle Measures in Degrees and
Radians

Figure 18 pg 9
10
P.5 Functions of Angles and Fundamental
Identities

• To define the six trigonometric functions, start
  with an angle ? in standard position. Choose any
  point P having coordinates (x,y) on the terminal
  side as seen in the figure below.
• Notice that r 0 since distance is never
  negative.

11
P.5 The Six Trigonometric Functions

• The six trigonometric functions are sine, cosine,
  tangent, cotangent, secant, and cosecant.

Trigonometric Functions Let (x,y) be a point
other than the origin on the terminal side of an
angle ? in standard position. The distance from
the point to the origin is
The six trigonometric functions of angle ? are
as follows.
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P.5 Finding Function Values of an Angle

• Example The terminal side of angle ? (beta) in
• standard position goes through (3,4). Find the
  values of the six trigonometric functions of ?.
• Solution

13
P.5 Reciprocal Identities

• Since sin? y/r and csc? r/y,
• Similarly, we have the following reciprocal
  identities for any angle ? that does not lead to
  a
• 0 denominator.

14
P.5 Using the Reciprocal Identities

• Example Find sin? if csc?
• Solution

15
P.5 Signs and Ranges of Function Values

• Example Identify the quadrant (or quadrants) of
  any angle ? that satisfies sin? 0, tan?
• Solution Since sin? 0 in quadrants I and II,
  while tan?
• in quadrants II and IV, both conditions are met
  only in quadrant II.

16
P.5 Signs and Ranges of Function Values

• Since
• for any angle ?.
• In a similar way,
• sec? and csc? are reciprocals of sin? and
  cos?, respectively, making

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P.5 Ranges of Trigonometric Functions

• Example Decide whether each statement is
  possible or
• impossible.
• (b) tan? 110.47 (c)
  sec? .6
• Solution
• Not possible since
• Possible since tangent can take on any value.
• Not possible since sec? ? 1 or sec? ? 1.

• For any angle ? for which for which the
  indicated function
• exists
• 1 ? sin? ? 1 and 1 ? cos? ? 1
• tan? and cot? may be equal to any real number
• sec? ? 1 or sec? ? 1 and csc? ? 1 or
  csc? ? 1.

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P.5 Pythagorean Identities

• Three new identities from x2 y2 r2
• Divide by r2
• Since cos? x/r and sin? y/r, this result
  becomes
• Divide by x2
• Dividing by y2 leads to cot2? 1 csc2?.

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P.5 Pythagorean Identities

• Example Find sin? and cos?, if tan? 4/3 and ?
  is in
• quadrant III.
• Solution Since ? is in quadrant III, sin? and
  cos? will
• both be negative.

Pythagorean Identities
20
P.5 Quotient Identities

• Recall that
  Consider the quotient of sin? and cos? where
  cos? ? 0.
• Similarly

Quotient Identities
21
Identities you should know
Even / Odd Properties
Compare the values of the trig function in QI and
QIV.
22
P.5 Evaluating Trigonometric Functions

• Acute angle A is drawn in
• standard position as shown.

Right-Triangle-Based Definitions of Trigonometric
Functions For any acute angle A in standard
position,
23
P.5 Finding Trigonometric Function Values of an
Acute Angle in a Right Triangle

• Example Find the values of sin A, cos A, and tan
  A in the right triangle.
• Solution
• length of side opposite angle A is 7
• length of side adjacent angle A is 24
• length of hypotenuse is 25

24
P.5 Trigonometric Function Values of Special
Angles

• Angles that deserve special study are 30º, 45º,
  and 60º.

Using the figures above, we have the exact values
of the special angles summarized in the table on
the right.
25
P.5 Cofunction Identities

• In a right triangle ABC, with right angle C, the
  acute angles A and B are complementary.
• Since angles A and B are complementary, and
• sin A cos B, the functions sine and cosine are
  called cofunctions. Similarly for secant and
  cosecant, and tangent and cotangent.

26
P.5 Cofunction Identities
If A is an acute angle measured in degrees,
then If A is an acute angle measured in
radians, then

• NoteThese identities actually apply to all angles
  (not just acute angles).

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Trig. Identities

• Review trig. identities on page

28
P.5 Reference Angles

• A reference angle for an angle ?, written ? ?, is
  the positive acute angle made by the terminal
  side of angle ? and the x-axis.
• Example Find the reference angle for each angle.
• 218º (b)
• Solution
• (a) ? ? 218º 180º 38º (b)

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P.5 Special Angles as Reference Angles

• Example Find the values of the trigonometric
• functions for 210º.
• Solution The reference angle for 210º is
• 210º 180º 30º.

Choose point P on the terminal side so that the
distance from the origin to P is 2. A 30º - 60º
right triangle is formed.
30
P.5 Finding Trigonometric Function Values Using
Reference Angles

• Example Find the exact value of each expression.
• cos(240º) (b) tan 675º
• Solution
• 240º is coterminal with 120º.
• The reference angle is
• 180º 120º 60º. Since 240º
• lies in quadrant II, the
• cos(240º) is negative.
• Similarly,
• tan 675º tan 315º
• tan 45º 1.

31
P.5 Finding Trigonometric Function Values with a
Calculator

• Example Approximate the value of each expression.
• cos 49º 12? (b) csc 197.977º
• Solution Set the calculator in degree mode.

32
P.5 Finding Angle Measure

• Example Using Inverse Trigonometric Functions to
  Find
• Angles
• Use a calculator to find an angle ? in degrees
  that satisfies sin ? ? .9677091705.
• Use a calculator to find an angle ? in radians
  that satisfies tan ? ? .25.
• Solution
• With the calculator in degree mode,
• we find that an angle having a sine
• value of .9677091705 is 75.4º. Write
• this as sin-1 .9677091705 ? 75.4º.
• With the calculator in radian mode,
• we find tan-1 .25 ? .2449786631.

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P.5 Finding Angle Measure

• Example Find all values of ?, if ? is in the
  interval 0º, 360º) and
• Solution Since cosine is negative, ? must lie in
  either quadrant II or III. Since
• So the reference angle ? ? 45º.
• The quadrant II angle ? 180º 45º 135º, and
  the
• quadrant III angle ? 180º 45º 225º.

34
P.5 Solving a Right Triangle Given an Angle and
a Side

• Example Solve the right triangle ABC, with A
  34º 30? and
• c 12.7 inches.
• Solution
• Angle B 90º A 89º 60? 34º 30?
• 55º 30?.
• Use given information to find b.

35
P.5 Solving a Right Triangle Given Two Sides

• Example Solve right triangle ABC if a 29.43
  centimeters and c 53.58 centimeters.
• Solution Draw a sketch showing the given
  information.
• Using the inverse sine function
• on a calculator, we find A ? 33.32º.
• B 90º 33.32º ? 56.68º
• Using the Pythagorean theorem,

36
P.5 Solving a Problem Involving Angle of Elevation

• Angles of Elevation or Depression
• Example Francisco needs to know the height of a
  tree. From a given point on the ground, he finds
  that the angle of elevation to the top of the
  tree is 36.7º. He then moves back 50 feet. From
  the second point, the angle of elevation is
  22.2º. Find the height of the tree.

37
P.5 Solving a Problem Involving Angle of Elevation

• Analytic Solution There are two unknowns, the
  distance x
• and h, the height of the tree.
• In triangle ABC,
• In triangle BCD,
• Each expression equals h, so the expressions
  must be equal.

38
P.5 Evaluating Circular Functions

• Example Evaluate and .
• Solution An angle of radians intersects the
  unit circle at the point (0, -1)

39
P.5 Special Angles and The Circular Functions
The special angles and their corresponding points
on the unit circle are summarized in the figure.
40
Review of Trigonometry
41
Review of Trigonometry
42

• Give in simplified radical form.

43

• Give in simplified radical form.

QII
QIII
QIV
QIII
44
P.5 Evaluating Circular Functions

• Example
• (a) Find the exact values of and
  .
• (b) Find the exact value of .

45
P.5 Evaluating Circular Functions

• Solution (a) From the figure
• The angle -5p/3 radians is coterminal with an
  angle of p/3 radians. From the figure

46
P.5 Graphs of the Sine and Cosine Functions

• Many things in daily life repeat with a
  predictable pattern. Because sine and cosine
  repeat their values over and over in a regular
  pattern, they are examples of periodic functions.

Periodic Function A periodic function is a
function f such that f (x) f (x np), for
every real number x in the domain of f , every
integer n, and some positive real number p. The
smallest possible positive value of p is the
period of the function.
47
P.5 Graph of the Sine Function

• From the graph we can see that as s
• increases, sin s oscillates between ?1.
• Using x rather than s, we can plot
• points to obtain the graph y sin x.
• The graph is continuous
• on (?,?).
• Its x-intercepts are of the
• form n?, n an integer.
• Its period is 2?.
• Its graph is symmetric
• with respect to the origin,
• and it is an odd function.

48
P.5 Graph of the Cosine Function

• The graph of y cos x can be found much the same
  way as
• y sin x.
• Note that the graph of y cos x is the graph y
  sin x translated units to the left.
• The graph is continuous
• on (?,?).
• Its x-intercepts are of the
• form (2n 1) , n an
• integer.
• Its period is 2?.
• Its graph is symmetric
• with respect to the y-axis,
• and it is an even function.

49
P.5 Graphing Techniques, Amplitude, and Period

• Example Graph y 2 sin x, and compare to the
• graph of y sin x.
• Solution From the table, the only change in the
• graph is the range, which becomes 2,2.

50
P.5 Amplitude

• The amplitude of a periodic function is half the
  difference between the maximum and minimum
  values.
• For sine and cosine, the amplitude is

Amplitude The graph of y a sin x or y a cos
x, with a ? 0, will have the same shape as y
sin x or y cos x, respectively, except with
range a, a. The amplitude is a.
51
P.5 Period

• To find the period of y sin bx or y cos bx,
  solve the inequality for b 0
• Thus, the period is
• Divide the interval into four equal parts to get
  the values for which y sin bx or y cos bx is
  1, 0, or 1. These values will give the minimum
  points, x-intercepts, and maximum points on the
  graph.

52
P.5 Graphing y cos bx

• Example Graph over one period.
• Analytic Solution

53
P.5 Graphing y cos bx

• Graphing Calculator Solution
• Use window 0,3? by 2,2, with Xscl 3?/4.
• Choose Xscl 3?/4 so that x-intercepts,
  maximums,
• and minimums coincide with tick marks on the axis.

54
P.5 Sketching Traditional Graphs of the Sine and
Cosine Functions

• To graph y a sin bx or y a cos bx, with b
  0,
• Find the period, Start at 0 on the x-axis,
  and lay off a distance of
• Divide the interval into four equal parts.
• Evaluate the function for each of the five
  x-values resulting from step 2. The points will
  be the maximum points, minimum points, and
  x-intercepts.
• Plot the points found in step 3, and join them
  with a sinusoidal curve with amplitude a.
• Draw additional cycles as needed.

55
P.5 Graphing y a sin bx

• Example Graph y 2 sin 3x.
• Solution
• Period
• Divide the interval into four equal
  parts to get the x-values

56
P.5 Graphing y a sin bx

• Plot the points (x, 2 sin 3x) from the table.
• Notice that when a is negative, the graph of
• y 2 sin 3x is a reflection across the x-axis
  of the graph of y a sin bx.

57
P.5 Translations

• Horizontal
• The graph of y f (x d) translates the graph
  of y f (x) d units to the right if d 0 and
  d units to the left
• if d
• A horizontal translation is called a phase shift
  and the expression x d is called the argument.
• Vertical
• The graph of y c f (x) translates the graph
  of y f (x) c units upward if c 0 and c
  units downward if c

58
P.5 Graphing y c a sin b(x d)

• Example Graph y 1 2 sin(4x ?).
• Solution Express y in the form c a sin b(x
  d).
• Amplitude 2
• Period
• Translate -1 1 unit downward
• and units to the left.
• Start the first period at x-value
  and end the first
• period at

59
P.5 Graphs of the Other Trigonometric Functions

• Graphs of the Cosecant and Secant Functions
• Cosecant values are reciprocals of the
  corresponding sine values.
• If sin x 1, the value of csc x is 1. Similarly,
  if sin x 1,
• then csc x 1.
• When 0 1. Similarly,
• if 1
• When approaches 0, the gets
  larger. The graph of y csc x approaches the
  vertical line x 0.
• In fact, the vertical asymptotes are the lines x
  n?.

60
P.5 Graphs of the Cosecant and Secant Functions

• A similar analysis for the secant function can be
  done. Plotting a few points, we have the solid
  lines representing the curves for the cosecant
  and secant functions.

61
P.5 Graphs of the Cosecant and Secant Functions

• Cosecant Function
• Discontinuous at values of x of the form x n?,
  and has vertical asymptotes at these values.
• No x-intercepts.
• Its period is 2? with no amplitude.
• Symmetric with respect to the origin, and is an
  odd function.
• Secant Function
• Discontinuous at values of x of the form (2n 1)
  , and has vertical asymptotes at these values.
• No x-intercepts.
• Its period is 2? with no amplitude.
• Symmetric with respect to the y-axis, and is an
  even function.

62
P.5 Sketching Traditional Graphs of the Cosecant
and Secant Functions

• To graph y a csc bx or y a sec bx, with b
  0,
• Graph the corresponding reciprocal function as a
  guide, using a dashed curve.
• Sketch the vertical asymptotes. They will have
  equations of the form x k, k an x-intercept of
  the guide function.
• Sketch the graph of the desired function by
  drawing the
• U-shaped branches between adjacent asymptotes.

To Graph
Use as a Guide
y a csc bx
y a sin bx
y a cos bx
y a sec bx
63
P.5 Graphing y a sec bx

• Example Graph
• Solution The guide function is
• One period of the graph lies along the interval
  that
• satisfies the inequality
• Dividing this interval into four equal parts
  gives the
• key points (0,2), (?,0), (2?,2), (3?,0), and
  (4?,2),
• which are joined with a smooth dashed curve.

64
P.5 Graphing y a sec bx

• Sketch vertical asymptotes where the guide
  function
• equals 0 and draw the U-shaped branches,
• approaching the asymptotes.

65
P.5 Graphs of Tangent and Cotangent Functions

• Tangent
• Its period is ? and it has no amplitude.
• Its values are 0 when sine values are 0, and
  undefined when cosine values are 0.
• As x goes from tangent values go
  from ? to ?, and increase throughout the
  interval.
• The x-intercepts are of the form x n?.

66
P.5 Graphs of Tangent and Cotangent Functions

• Cotangent
• Its period is ? and it has no amplitude.
• Its values are 0 when cosine values are 0, and
  undefined when sine values are 0.
• As x goes from 0 to ?, cotangent values go from ?
  to
• ?, and decrease throughout the interval.
• The x-intercepts are of the form x (2n 1)

67
P.5 Sketching Traditional Graphs of the Tangent
and Cotangent Functions

• To graph y a tan bx or y a cot bx, with b
  0,
• The period is To locate two adjacent
  vertical asymptotes, solve the following
  equations for x
• Sketch the two vertical asymptotes found in Step
  1.
• Divide the interval formed by the vertical
  asymptotes into four equal parts.
• Evaluate the function for the first-quarter
  point, midpoint, and third-quarter point, using
  x-values from Step 3.
• Join the points with a smooth curve approaching
  the vertical asymptotes.

For y a tan bx bx and bx
For y a cot bx bx 0 and bx ?.
68
P.5 Graphing y a cot bx

• Example Graph
• Solution Since the function involves cotangent,
  we can
• locate two adjacent asymptotes by solving the
  equations
• Dividing the interval
• into four equal parts and finding
• the key points, we get

69
Review of Trigonometry

• Amplitude
• Period
• Phase shift
• Vertical shift

70
Review of Trigonometry

• Amplitude
• Period
• Phase shift
• Vertical shift

Height of each wave.
Length of one wave.
Re-center the graph at x C.
Re-center the graph at y D.
71

• Create these graphs by making the sine or cosine
  graphs first and then inverting individual
  points.
• Amplitude
• Period
• Phase shift
• Vertical shift

72

• Create these graphs by making the sine or cosine
  graphs first and then inverting individual
  points.
• Amplitude
• Period
• Phase shift
• Vertical shift

Only applies to sine and cosine.
All are the same as sine and cosine
73

• Amplitude
• Period
• Phase shift, Vertical shift

74

• Amplitude
• Doesnt really apply (not a wave)
• Gives a point the graph passes through
• Period
• Repeats twice as fast as sin, cos, sec, csc
• Phase shift, Vertical shift

Same as sin, cos, sec, csc
75
Graph one period, beginning at zero.

• Amplitude
• Period
• Phase shift
• Vertical shift

76
Graph one period, beginning at zero.

• Amplitude
• Period
• Phase shift
• Vertical shift

Re-center graph at
77

• Function Domain Range

78
ysin x
All the solutions for x can be expressed in the
form of a general solution.
79
Example General Solution
Find the general solution for the equation sec ?
2.
80
Example Solve tan x1
Example Solve tan x 1.
The graph of y 1 intersects the graph of y
tan x infinitely many times.
81
Example Solve the Equation

82
Example Find all solutions using unit circle
83
Example Find all solutions
Find all solutions of the trigonometric equation
tan2 ? tan ? 0.
Therefore, tan ? 0 or tan ? -1.
The solutions for tan ? 0 are the values ?
kp, for k any integer.
84
Quadratic Form
The trigonometric equation 2 sin2 ? 3 sin ? 1
0 is quadratic in form.
2 sin2 ? 3 sin ? 1 0 implies that
(2 sin ? 1)(sin ? 1) 0.
Therefore, 2 sin ? 1 0 or sin ? 1 0.
? -p 2kp, from sin ? -1
x
85
Example Solutions in an interval
Example Solve 8 sin ? 3 cos2 ? with ? in the
interval 0, 2p.
Rewrite the equation in terms of only one
trigonometric function.
Therefore, 3 sin ? ? 1 0 or sin ? 3 0
s
86
Example Solve quadratic equation
Solve 5cos2 ? cos ? 3 0 for 0 ? p.
The equation is quadratic. Let y cos ? and
solve 5y2 y ? 3 0.
Therefore, cos ? 0.6810249 or 0.8810249.
Use the calculator to find values of ? in 0 ?
p.
This is the range of the inverse cosine function.
The solutions are ? cos ?1(0.6810249 )
0.8216349 and ? cos ?1(?0.8810249) 2.6488206
87

• Solve on the interval .

88

• Solve on the interval .