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BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§8.1 Angles &
TrigoNometry
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §7.6 → Double Integrals
Any QUESTIONS About HomeWork• §7.6 → HW-9
7.6
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 3
Bruce Mayer, PE Chabot College Mathematics
Angles: Basic Terms
Two distinct points determine a line called Line AB
Line segment AB → a portion of the line between A and B, including points A and B.
Ray AB → a portion of line AB that starts at A and continues through B, and on past B
A B
A B
A B
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 4
Bruce Mayer, PE Chabot College Mathematics
Angles: Basic Terms Angle: formed by
rotating a ray around its endpoint.
The ray in its initial position is called the initial side of the angle
The ray in its location after the rotation is the terminal side of the angle
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 5
Bruce Mayer, PE Chabot College Mathematics
Identifying Angles Unless it is ambiguous as to the meaning, angles
may be named only by a single letter (English or Greek) displayed at vertex or in area of rotation between initial and terminal sides
Angles may also be named by three letters, one representing a point on the initial side, one representing the vertex and one representing a point on the terminal side (vertex letter in the middle, others first or last)B
c
:Names AcceptableA angle
CAB angleBAC angle
angle
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 6
Bruce Mayer, PE Chabot College Mathematics
Positive & Negative Angles Positive angle: The
rotation of the terminal side of an angle counterclockwise.
Negative angle: The rotation of the terminal side is clockwise.
Positive Angle Negative Angle
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 7
Bruce Mayer, PE Chabot College Mathematics
Angle: Measures & Classes
The most common unit for measuring angles is the degree (°) • One Rotation or Cycle = 360°
Four Classes of Angle:• Acute, Right, Obtuse, Straight
oo 900 oo 18090 o90 o180
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 8
Bruce Mayer, PE Chabot College Mathematics
Angle: RADIAN Measure
Define the “Radian” measure as the SubTended Circumferential distance on a circle divided by the radius.
Thus a subtendedangle that produces anarc-length of 1 radius is 1 radian in measure
Radians inone Cycle:
22
Radius Circle
nceCircumfere Circle
r
r
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 9
Bruce Mayer, PE Chabot College Mathematics
Degrees & Radians ComparedMeasure Description Graphic
One Quarter Revolution
One HalfRevolution
Three Quarter Revolution
One FullRevolution
90radians 2
180radians
270radians 2
3
360radians 2
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 10
Bruce Mayer, PE Chabot College Mathematics
Degrees ↔ Radians
The Measure of One Cycle
Then the Number “1”
Convert to other Measure: 53°, 2.2 rad
rads 2Cycle One360
rads
180
rads 2
360
Cycle One
Cycle One
rad 925.0180
rads 53153
05.126rads
180rad 2.21rad 2.2
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 11
Bruce Mayer, PE Chabot College Mathematics
Unit Circle Imagine a circle on
the CoOrdinate plane, with its center at the origin, and a radius of 1.
Choose a point on the circle somewhere in quadrant I.
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 12
Bruce Mayer, PE Chabot College Mathematics
Unit Circle Connect the origin
to the point, and from that point drop a perpendicular to the x-axis.
This creates a right triangle with hypotenuse of 1.
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 13
Bruce Mayer, PE Chabot College Mathematics
Unit Circle The length of its legs
are the x and y coordinates of the chosen point.
Applying the definitions of the trigonometric ratios to this triangle gives
x
y1
is the angle of rotation
xx
1Hyp
Adjcosy
y
1Hyp
Oppsin
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 14
Bruce Mayer, PE Chabot College Mathematics
Unit Circle
Thus The CoOrdinates of the chosen point are the CoSine (x) and Sine (y) of the angle • This provides a way to define functions
sin() and cos() for all real numbers
• The Four other trigonometric functions can be defined from the Unit Circle as well
yx sincos
y
xx
y
Opp
Adjcot
Adj
Opptan
y
x1
Opp
Hypcsc
1
Adj
Hypsec
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 15
Bruce Mayer, PE Chabot College Mathematics
The 16-Point Unit Circle
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 16
Bruce Mayer, PE Chabot College Mathematics
Un
it Circ Tab
ulated
0
1/2
2 /2
3 /2
1
1
3 /2
2 /2
1/2
0
0
3 /3
1
3
2
2
2 3 /3
1
1
2 3 /3
2
2
3
1
3 /3
0
3 /2
2 /2
1/2
0
1/2
2 /2
3 /2
1
3
1
3 /3
0
2 3 /3
2
2
2
2
2 3 /3
1
3 /3
1
3
1/2
2 /2
3 /2
1
3 /2
2 /2
1/2
0
3 /2
2 /2
1/2
0
1/2
2 /2
3 /2
1
3 /3
1
3
3
1
3 /3
0
2
2
2 3 /3
1
2 3 /3
2
2
2 3 /3
2
2
2
2
2 3 /3
1
3
1
3 /3
0
3 /3
1
3
cossin tan csc sec cot
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 17
Bruce Mayer, PE Chabot College Mathematics
Example Calc Sin & CoSin Find the
values: Negative angles are represented by
traversing the Unit Circle ClockWise, so the terminal side of an angle of −π/2 rads (−90°) falls on the negative y-axis and takes the point (1,0) to the point (0,−1).
The CoSine is given by the x-coordinate at this point, so
45
2 sin and cos vu
0cos 2 u
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 18
Bruce Mayer, PE Chabot College Mathematics
Example Calc Sin & CoSin
SOLUTION: The terminal side of the angle with
measure 5π/4 rads (225°) falls on the line in the third quadrant which takes the point (1,0) to the point:
The Sine is the y-coordinate of this point, so
45sin v
22
22 ,sin,cos, yx
22sin 45 vv
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 19
Bruce Mayer, PE Chabot College Mathematics
Graph: Sine & CoSine
-10 -5 0 5 10-1
-0.5
0
0.5
1
x
y =
sin
()
MTH16 • sin()
-10 -5 0 5 10-1
-0.5
0
0.5
1
x
y =
co
s( )
MTH16 • cos()
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 20
Bruce Mayer, PE Chabot College Mathematics
Properties of Sine & CoSine From the Periodic
Nature of the Sinusoidal Graphs Observe
-10 -5 0 5 10-1
-0.5
0
0.5
1
x
y =
sin
()
MTH16 • sin()
-10 -5 0 5 10-1
-0.5
0
0.5
1
x
y =
co
s( )
MTH16 • cos()
coscos
sinsin
cos2cos
sin2sin
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 21
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-16 • 22Feb14% MTH15_Quick_Plot_BlueGreenBkGnd_130911.m%clear; clc; clf; % clf clears figure window%% The Domain Limitsxmin = -4*pi; xmax = 4*pi;% The FUNCTION **************************************x = linspace(xmin,xmax,1000); y = sin(x); y1 = cos(x);% ***************************************************% the Plotting Range = 1.05*FcnRangeymin = min(y); ymax = max(y); % the Range LimitsR = ymax - ymin; ymid = (ymax + ymin)/2;ypmin = ymid - 1.025*R/2; ypmax = ymid + 1.025*R/2% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greensubplot(2,1,1)plot(x,y, 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = sin(\theta)'),... title(['\fontsize{16}MTH16 • sin(\theta)'])hold onplot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)hold offsubplot(2,1,2)plot(x,y1, 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = cos(\theta)'),... title(['\fontsize{16}MTH16 • cos(\theta)',])hold onplot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)hold off
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 22
Bruce Mayer, PE Chabot College Mathematics
Trig Fcn RelationShips
4 of the 6 Trig Functions can be expressed in Terms of the basis functions of sin and cos
With reference to the Unit Circle Find
sin
coscot
cos
sintan
y
x
x
y
sin
11csc
cos
11sec
yx
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 23
Bruce Mayer, PE Chabot College Mathematics
Pythagorean Identities ReCall the
Pythagorean Theorem
The Unit Circle Analogy
sin,cos
1sincos1 22222 yx
yx,
22
2
2
2
22
2
2
2
2
2
222
sec11
111tan
xxx
xy
x
x
x
y
x
y
x
y
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 24
Bruce Mayer, PE Chabot College Mathematics
Pythagorean Identities Also
In Summary
sin,cos yx,
2
2
2
2
2
22
2
2
2
2
2
22
2
csc11
11cot1
yyy
xy
y
x
y
y
y
x
y
x
22
22
22
csccot1
sec1tan
1cossin
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 25
Bruce Mayer, PE Chabot College Mathematics
Example Use Trig Relns
Find the value of cos(θ) given that• csc(θ) = 3 • the angle θ is contained in a right triangle
SOLUTION: Recall from Unit Circle:
Next use the Pythagorean Identity3
1sin
31
13csc
sin
11csc
y
2222 311cossin1cos
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 26
Bruce Mayer, PE Chabot College Mathematics
Example Use Trig Relns
Then in This case
So
But since θ is confined to right triangle θ must be less than 90° then the cos must be POSITIVE
Thus if csc(θ) = 3, then
9428.03
22cos
9
8
9
1
9
9cos2
3
22
3
24
3
24
9
8
9
8
9
8cos2
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 27
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
A math Model for the Diurnal hours of daylight t months after January 1 in Eugene, Oregon
Use this model to • Find the amplitude, period, horizontal
and vertical shifts of the function. – Interpret the values
2.1251.154.0sin17.3 ttD
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 28
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
SOLUTION: The amplitude is the distance from
average to high (or average to low) values of the function. This is represented by the absolute value of the CoEfficient on the trigonometric function (sine in this case).
2.1251.154.0sin17.3 ttD
amplitude
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 29
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
Thus by the sinusoidal amplitude over time, the daylight hours in Eugene varies 3.17 up & down from its average.
SOLUTION: The period of a sine function is the
value p when written in the form
dx
pbaxf
2sin
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 30
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
Factor to produce a t-CoEfficient of this form in the given function-argument:
Then by sine-argument Correspondence
The function repeats itself every 11.64 months, which is probably a rough approximation of the 12-month yearly cycle of daylight
80.254.051.154.0 tt
64.1154.0
2254.0
p
p
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 31
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
SOLUTION: The horizontal shift (also called the
Phase-Shift) of the function is given by the value of d in the form
Again by sine-argument Correspondence
dx
pbaxf
2sin
80.280.2 ddxt
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 32
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
The d = 2.8 months suggests that the average value is not achieved at t = 0 (December 31st), but rather the function is close to its minimum in early spring, about 2.8 months in to the Year.
SOLUTION: The vertical shift (also called the mean
value) of the function is given by the value of a in the form
Again by sine-argument Correspondence
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 33
Bruce Mayer, PE Chabot College Mathematics
Example Sinusoidal Periodicity
Then by function Correspondence
The function does not vary equally above and below zero (negative daylight hours makes no sense). Instead, the average value is 12.2 hours and the function varies up and down from that midline.
2.122.1251.154.0sin17.32
sin
atadx
pb
dx
pbaxf
2sin
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 34
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §8.1• P8.1-68 →
HomeHeating EnergyUse in Buffalo, NewYork
2010 Weather Summary Buffalo, NY New York USA
Weather Index 22 29 100
Hail Index 3 39 100
Hurricane Index 34 93 100
Tornado Index 40 33 100
Annual Maximum Avg. Temperature 56.0 °F 57.0 °F N/A
Annual Minimum Avg. Temperature 40.0 °F 39.0 °F N/A
Annual Avg. Temperature 47.7 °F 47.7 °F N/A
Annual Heating Degree Days (Tot Degrees < 65) 6,747 6,762 N/A
Annual Cooling Degree Days (Tot Degrees > 65) 477 484 N/A
Percent of Possible Sunshine 48 51 N/A
Mean Sky Cover (Sunrise to Sunset - Out of 10) 7 7 N/A
Mean Number of Days Clear (Out of 365 Days) 54 65 N/A
Mean Number of Days Rain (Out of 365 Days) 169 150 N/A
Mean Number of Days Snow (Out of 365 Days) 26 21 N/A
Avg. Annual Precipitation (Total Inches) 39.00" 38.00" N/A
Avg. Annual Snowfall (Total Inches) 91.00" 75.00" N/A
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 35
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
MoreTrig
Identities
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 36
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
–
srsrsr 22
a2 b2
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 37
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 38
Bruce Mayer, PE Chabot College Mathematics
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