[email protected] mth16_lec-09_sec_7-6_double_integrals.pptx 1 bruce mayer, pe chabot college...

[email protected] • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§8.1 Angles &

TrigoNometry



Review §

Any QUESTIONS About• §7.6 → Double Integrals

Any QUESTIONS About HomeWork• §7.6 → HW-9

7.6

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=3O7KQ1HrjwzJSM&tbnid=DrhGRO_1dx_RFM:&ved=0CAUQjRw&url=http://www.imomath.com/index.php?options=470&lmm=0&ei=KTIJU6O8AYr9oAS28oCIAg&bvm=bv.61725948,d.cGU&psig=AFQjCNEcO_7F9VJYs04wBkv8mEmNpabj3g&ust=1393197823611938



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



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



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



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



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



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



Degrees & Radians ComparedMeasure Description Graphic

One Quarter Revolution

One HalfRevolution

Three Quarter Revolution

One FullRevolution

90radians 2

180radians

270radians 2

3

360radians 2



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



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.



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.



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



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



The 16-Point Unit Circle



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



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



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



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()



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



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



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



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

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=xwjLcTdb2zHhEM&tbnid=J85pDJsopWVxjM:&ved=0CAUQjRw&url=http://www.algebra-class.com/basic-algebra-formulas.html&ei=LW8JU_CIL47eoATov4GoBA&psig=AFQjCNHKy02OgghRkT0x2pWr3_N9Bw-_kw&ust=1393213600764524



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



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



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



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




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




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




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




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




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




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



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



All Done for Today

MoreTrig

Identities

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=36QefxUlMAjl9M&tbnid=I89vUC-fN_VV9M:&ved=0CAUQjRw&url=http://www.artofmanliness.com/2012/11/19/how-throw-a-spiral-football/&ei=Io7VUrvACY-hogSr_IKoDA&bvm=bv.59378465,d.cGU&psig=AFQjCNFpdAGRelnFsLiPrVTkxg_YeEefqw&ust=1389813543017809



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22

a2 b2

[email protected] mth16_lec-09_sec_7-6_double_integrals.pptx 1 bruce mayer, pe chabot college...

Documents

chabot mathematics

b slide

revolution slide

angles trigonometry

classes of angle

subtended angle

rad slide

straight slide