Trig Calculus

7/31/2019 Trig Calculus

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Retnodh_Rossana 2010


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Angle Measurement

To convert from degrees to radians, multiply by .180

To convert from radians to degrees, multiply by.

180

radians, so radians2360 180


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Special Angles

0

902

180

1354

3

1203

2

1506

5 306

454

603

r=1

27023 /


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Special Angles - Unit Circle Coordinates

0,1

21,2

3

21,

21

2321 ,1,0

23,

21

21,

21

21,23

0,1

r=1

/3

5/6/4

/2

2/33/4

/6

0

3/2

10,


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Trig Functions - Definitions

r

ysin

r

xcos

x

ytan

y

rcsc

x

rsec

y

xcot

22 yxr

(x,y)r


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Trig Functions - Definitions

hyp

adjcos

hyp

oppsin

opp

adj

tan

1cot

opp

hyp

sin

1csc

adj

hyp

cos

1sec

cos

sintan

adj

opp

opp

adj

hyp


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Trig Functions

Signs by quadrants

all functions positivesin, csc positive

tan, cot positive cos, sec positive


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Special Angles - Triangles

454

45

1

1

2

hypoppsin

4

example:

2221


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Special Angles - Triangles

306

603

2

1

3

2

160cos

hyp

adj

3

3

3

130tan

adj

opp

23,

21

r=1 21

23

,

2

1

1

21

60cos r

x

3

3

23

21

30tan x

y


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Special Angles

For the angles

23270,180,

290,0

example:

2

3sin

r

y1

1

1

Use the unit circle points (1,0), (0,1), (-1,0) and (0,-1) or look at

the graphs for the trig functions

r = 1

(1,0)

(0,1)

(0,-1)

(-1,0)


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Graphing Trigonometry Functions

Basic Graphs

y = sin x

1

-1

-/2 /2 3/2 2

Period is and amplitude is 1.2


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Graphing Trigonometry Functions

Basic Graphs

-/2 /2 3/2 2

y = cos x1

-1



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Special Angles and Graphs

1

1

Using the graph for cos,xcosy 1


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Graphing Trig FunctionsAmplitude Change

y=asin x stretches or compresses the graph vertically

y =asin x

a

-a

-/2 /2 3/2 2

Period is and amplitude isa.2


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Graphing Trig FunctionsPhase Shift

y = sin(x -b) slides graph right byb units

b

-1

1

2+b

y = sin(x -b)



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Graphing Trig FunctionsPhase Shift

y = sin(x +b) slides graph left byb units

1

-b 2- b

y = sin(x +b)

-1



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Graphing Trig FunctionsPeriod Change

y = sincx stretches or compresses the graph horizontally

-1

1

2/c

Period is and amplitude is 1.c/2


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The rules for shifting, stretching, shrinking, and reflecting the

graph of a function apply to trigonometric functions.

y a f b x c d

Vertical stretch or shrink;reflection aboutx-axis

Horizontal stretch or shrink;

reflection abouty-axis

Horizontal shift;

Vertical shift;

Positive c moves left.

Positive dmoves up.

The horizontal changes happen in

the opposite direction to what you

might expect.

is a stretch.1a

is a shrink.1b


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0

1

2

3

4

-1 1 2 3 4 5

x

When we apply these rules to sine and cosine, we use some

different terms.

2

sinf x A x C DB

Horizontal shift

Vertical shift

is the amplitude.A

is the period.B

A

B

C

D 2

1.5sin 1 24

y x


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Trig Identities

Reciprocal Quotient

cossec

1

tancot

1

sincsc

1

sin

cos

cot

cossintan


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Trig Identities

Pythagorean

1cossin 22

22 sec1tan

22 csccot1

Double Angle

cossin22sin

2

2

22

sin21

1cos2

sincos2cos


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Trig Identities

Sum and difference

BsinAcosBcosAsinBAsin

BsinAcosBcosAsinBAsin

BsinAsinBcosAcosBAcos

BsinAsinBcosAcosBAcos


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Inverse Trig Functions

is equivalent toxarcsinxsiny1

ysinx

is equivalent toxarccosxcosy1

ycosx


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Solving Trig Equations

Use algebra, then inverse trig functions or knowledgeof special angles to solve.

202

1sin

example: if

0sin in quadrants I and II

and since6

5

62

1arcsin

2

1sin 1


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Line of Sight

Horizontal


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Angel of Elevation

Angel ofElevation


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Angle of Depression

Horizontal

Angel of

Depression


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The angle of elevation of the top of a tower from a

point on the ground, which is 30 m away from the foot

of the tower is 30. Find the height of the tower.

Let AB be the tower and the angle of elevation from

point C (on ground) is 30.

ABC,

.

Therefore, the height of the tower is


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A kite is flying at a height of 60 m above the ground.

The string attached to the kite is temporarily tied to a

point on the ground. The inclination of the string withthe ground is 60. Find the length of the string,

assuming that there is no slack in the string.

Let K be the kite and the string is tied to point P on

the ground.

KLP,

.

Hence, the length of the string is


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Applications of Trigonometry: mechanical engineering, civil engineering, architecture,

electrical engineering, electronics, land surveying andgeodesy, many physical sciences, analysis of financialmarkets, economics, medical imaging (CAT scans andultrasound), probability theory, statistics, biology,chemistry, seismology, meteorology, oceanography,computer graphics, cartography, crystallography and

game development, astronomy, navigation, and on.

31


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POLAR COORDINATE

r

)B(r, Koordinat Kutub

B(r, )


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CARTESIAN COORDINATE

Koordinat kartesius

A (x,y)

y)A(x,


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MENGUBAH KOORDINAT KUTUBMENJADI KOORDINAT KARTESIUS

Koordinat kutub B(r,)

dari diperoleh x = r . cos

dari diperoleh y = r . sin

sehingga didapat Koordinat kartesius

B(x,y) = B(r.Cos , r.Sin)

Cos

r

x

Sinr

y


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MENGUBAH KOORDINAT KARTESIUSMENJADI KOORDINAT KUTUB

Koordinat kartesius A (x,y)

22

yxr

x

ytan

x

yarc.tan

sehingga koordinat kutub A (r,


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bCDsin a.sinBCD

b.sinACD

a

CDsinB

b.sinAa.sinB

SinBb

SinAa


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ATURAN SINUS DAN COSINUS

Aturan Sinus:

C

c

B

b

A

a

sinsinsin

Aturan Cosinus:


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CONTOH SOAL :

Pada segitiga ABC, diketahui c = 6, sudut B = 600

dan sudut C = 450. Tentukan panjang b !Jawab:

0

26

3

45

6

60

21

21

00

b

SinSin

b

SinC

c

SinB

b

632

66

2

2

2

36

2

63

21

21

b

b

b


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Pada segitiga ABC, diketahui a = 6, b = 4 dan

sudut C = 1200 Tentukan panjang c !

Jawab:

c2 = a2 + b2 2.a.b.cos C

c2 = (6)2 + (4)2 2.(6).(4).cos 1200c2 = 36 + 16 2.(6).(4).( )

c2 = 52 + 24

c2

= 76c =76 = 219

Trig Calculus

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