4.4 Trigonometric Functions of Any Angle€¦ · 1 4.4 Trigonometric Functions of Any Angle r-r-r r...
Transcript of 4.4 Trigonometric Functions of Any Angle€¦ · 1 4.4 Trigonometric Functions of Any Angle r-r-r r...
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4.4 Trigonometric Functions of Any Angle
r
-r
-r
r
y
x
every point on the circle satisfies to x2+y2=r2
is any angle in standard position
any circle of radius r
P(x,y)
P(x,y) is any point onthe terminal side of
y
x
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4.4 Trigonometric Functions of Any Angle
r
-r
-r
r
y
x
P(x,y)y
x
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
cscθ=ry
secθ=rx
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4.4 Trigonometric Functions of Any Angle
r
-r
-r
r
y
x
P(x,y)y
x
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
cscθ=ry
secθ=rx
+ +
- -
- +
- +
- +
+ -
+ +
- -
- +
- +
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4.4 Trigonometric Functions of Any Angle
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
+ +
- -
- +
- +
- +
+ -
Examples:1) point (-12,5) lies on the terminal side of standard angle . Find the exact values of sin , cos , tan and cot .
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4.4 Trigonometric Functions of Any Angle
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
+ +
- -
- +
- +
- +
+ -
Examples:1) point (-12,5) lies on the terminal side of standard angle . Find the exact values of sin , cos , tan and cot .
Solution:(1) find the radius:r
r
y
x
(−12,5)
sinα=yr=
−1213
-125
52+(−12)2=r2
25+144=r2
r2=169 r=13(2)
cosα=xr=513
tanα=yx=−
512
cotα=−125
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4.4 Trigonometric Functions of Any Angle
tanθ=yx
cot θ=xy
secθ=rx
- +
+ -
- +
- +
Examples:2) Find tan
3π2, sec π ,cot π
2
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4.4 Trigonometric Functions of Any Angle
tanθ=yx
cot θ=xy
secθ=rx
- +
+ -
- +
- +
Examples:2) Find
Solution:(1) use the unit circle
(radius = 1)
(2)
(0,1)
1
y
x
tan3π2, sec π ,cot π
2
π2
3 π2
π(-1,0)
(0,-1)
tan3π2
=yx=
−10
=undefined
sec π=rx=1
−1=−1
cot π2=xy=01=0
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4.4 Trigonometric Functions of Any Angle
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
cscθ=ry
secθ=rx
+ +
- -
- +
- +
- +
+ -
+ +
- -
- +
- +
Examples:3) Given , in quadrant II,
find the exact value of each of the remaining trigonometric functions of .
cosβ=−35
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4.4 Trigonometric Functions of Any Angle
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
cscθ=ry
secθ=rx
+ +
- -
- +
- +
- +
+ -
+ +
- -
- +
- +
Examples:3) Given , in quadrant II,
find the exact value of each of the remaining trigonometric functions of .
5
cosβ=−35
-3
Solution:(1) find the radius
(2)
52=(−3)2+ y2y
y2=16 y=4
sinβ=yr=45
tanβ=yx=−
43
cotβ=−34
cscβ=ry=54
secβ=rx=−
53
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4.4 Trigonometric Functions of Any Angle
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
cscθ=ry
secθ=rx
+ +
- -
- +
- +
- +
+ -
+ +
- -
- +
- +
Examples:4)
find the exact value of each of the remaining trigonometric functions of .
tanθ=512
,cosθ<0
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4.4 Trigonometric Functions of Any Angle
sinθ=yr
cosθ=xr
tanθ=yx
cot θ=xy
cscθ=ry
secθ=rx
+ +
- -
- +
- +
- +
+ -
+ +
- -
- +
- +
Examples:4)
find the exact value of each of the remaining trigonometric functions of .
tanθ=512
Solution:(1) is in the III quadrant, point (-12,-5)(2) the radius is:
(3)
(−5)2+(−12)2=r2
y r2=169 r=13
sinθ=yr=
−513
cot θ=125
cscθ=ry=−
135
secθ=rx=−
1312
,cosθ<0
(−12 ,−5) cosθ=xr=
−1213
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4.4 Trigonometric Functions of Any Angle
y
x
Reference Angles
If is a non-acute angle in standard position that lies in a quadrant, then its reference angle is a positive angle ’ formed by the terminal side of and x-axis.
y
x
y
x
’
’’
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4.4 Trigonometric Functions of Any Angle
Reference Angles
If is a non-acute angle in standard position that lies in a quadrant, then its reference angle is a positive angle ’ formed by the terminal side of and x-axis.
The values of trigonometric functions of a given angle are the same as the values of the trigonometric functions of the reference angle ’, except possibly for the sign.
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4.4 Trigonometric Functions of Any Angle
Examples: use reference angles and trigonometric functions of special values table to evaluate
1)
2)
3)
sin4π3
cot(−π4 )
cos135o- +
- +
+ +
- -
- +
+ -
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4.4 Trigonometric Functions of Any Angle
Examples: use reference angles and trigonometric functions of special values table to evaluate
1) ’ = 180º – 135º = 45º,
2)
3)
sin4π3
cot(−π4 )
cos135o- +
- +
+ +
- -
- +
+ -
cos135o=−cos45o=−√22
θ '=4π3
−π=π3
, is in the III quadrant
sin4π3
=−sin π3=−
√32
, is in the II quadrant
, is in the IV quadrant + cot is odd f.
θ '=π4
cot(−π4 )=−cot(
π4 )=−1
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4.4 Trigonometric Functions of Any Angle
Examples: use reference angles and trigonometric functions of special values table to evaluate
4)
5)
6)
sec240o
cos7π4
sin14π3
- +
- +
- +
- +
+ +
- -
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4.4 Trigonometric Functions of Any Angle
Examples: use reference angles and trigonometric functions of special values table to evaluate
4)
5)
6)
sec240o
cos7π4
sin14π3
- +
- +
- +
- +
+ +
- -, is in the II quadrant
14 π3
=4 π+2π3
θ '=π2π3
=π3
sin14π3
=sin π3=
√32
, is in the III quadrant
’ = 240º – 180º = 60º, sec240o=−sec60o=−2
7 π4
=π+3π4
θ '=2π−7π4
=π4
, is in the IV quadrant
cos7π4
=cos π4=
√22