Download - 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
Period is and amplitude is 1.2
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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)
Period is and amplitude is 1.2
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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
Period is and amplitude is 1.2
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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.
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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