chapter 4.3 part 1 circular functions.pdf
TRANSCRIPT
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Chapter Chapter 4.3 A4.3 ACircular FunctionsCircular Functions
Sine FunctionSine Function
Let be the wrapping function. Then if
sine functio
, ,
the is defn ined assin .
PP t x y
t y
Cosine FunctionCosine Function
Let be the wrapping function. Then if,
cosine functi
,
the is defined as.
oncos
PP t x y
t x
Example 4.3.1Example 4.3.1
Evaluate the following.1. sin 2 0 cos 2 1
2 1,0
2 3 2 12. sin cos
3 2 3 2
2 1 3,
3 2 2
P
P
7 3 7 13. cos sin6 2 6 2
7 3 1,6 2 2
P
Tangent and Cotangent FunctionsTangent and Cotangent Functions
Let be the wrapping function. Then if, ,
then
tan , 0
cot , 0
PP t x y
yt xxxt yy
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Secant and Cosecant FunctionsSecant and Cosecant Functions
Let be the wrapping function. Then if, ,
then1sec , 0
1csc , 0
PP t x y
t xx
t yy
,
sin cos
tan , 0 cot , 0
1 1sec , 0 csc , 0
P t x y
t y t xy xt x t yx y
t x t yx y
Example 4.3.2Example 4.3.2Find the 6 circular function values when .
331. sin
3 212. cos
3 2
323. tan 313 2
1 1 324. cot3 33 3
2
t
1 3,
3 2 2P
Example 4.3.2Example 4.3.2
Find the 6 circular function values when .3
t
1 3,
3 2 2P
5. sec 2
3
2 2 36. csc3 33
Example 4.3.3Example 4.3.3
Find the 6 circular function values when .1. sin 0
2. cos 10
3. tan 01
4. cot is undefined15. sec 11
6. csc is undefined
t
1,0P
Challenge!Challenge!
Evaluate the following as fast as you can.2 1 41. cos 5. sec 23 2 3
2. tan 1 6. csc 0 is undefined4
11 1 5 23. sin 7. sin6 2 4 2
34. cot 3 8. tan6 6 3
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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-2
-1
1
2
t
f (t)
x
y
Domain, Range, and GraphsDomain, Range, and Graphs
1. sin
1,1
f t t
Dom f
Rng f
,
sin
P t x y
t y
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-2
-1
1
2
t
f (t)
x
y
Domain, Range, and GraphsDomain, Range, and Graphs
2. cos
1,1
f t t
Dom f
Rng f
,
cos
P t x y
t x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
t
f (t)
Domain, Range, and GraphsDomain, Range, and Graphs
3. tan
, is an odd integer2
f t t
kDom f t t k
Rng f
, tan yP t x y t x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
t
f (t)
Domain, Range, and GraphsDomain, Range, and Graphs
4. cot
, is an integer
f t t
Dom f t t k k
Rng f
, cot xP t x y t y
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
t
f (t)
Domain, Range, and GraphsDomain, Range, and Graphs
5. sec
, is an odd integer2
, 1 1,
f t t
kDom f t t k
Rng f
1, secP t x y t x
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
t
f (t)
Domain, Range, and GraphsDomain, Range, and Graphs
6. csc
, is an integer
, 1 1,
f t t
Dom f t t k k
Rng f
1, cscP t x y t y
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Periodic FunctionsPeriodic Functions
are functions whose valuesrepeat after a fixed interval.
The fixed interval is c
Periodic
alled the
functions
pe riod.-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12
-2
-1
1
2
x
f (t)
Periodic FunctionsPeriodic Functions
The six circular functions are periodic.
period
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12
-2
-1
1
2
x
f (t)
CycleCycle
A is simply the portion of a graph of afunction over an interval of length equalto the per
cycle
iod.
1 cycle 1 cycle 1 cycle
AmplitudeAmplitude
The of a graph of the sine or cosine function refers to one-half the absolute difference between the highest and the lowest function
amplitud
valu
e
es.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-2
-1
1
2
t
f (t)
AmplitudeAmplitude
1 1amplitude 1 1 2 12 2
Sine and Cosine FunctionsSine and Cosine Functions
Form: sin
cosDomain:
Range: ,
2Amplitude: Period:
f x a b x h k
g x a b x h k
k a k a
ab