Lecture 1.41 Value of Trig Functions
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Transcript of Lecture 1.41 Value of Trig Functions
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 1/52
0.2: Trignometric FunctionsObjectiv
e:•Understandrelationships between
angles and sides of righttriangles
•Review trig functions of ©2008 Roy L. Gover(www.mrgover.com)
Calculus PrerequisiteKnowledge
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 2/52
Definition
The special angles are:•30°,60°, and 45° in
degrees
• and in radians
, ,
6 3 4
π π π
In calculus, we use
radians.
8/7/2019 Lecture 1.41 Value of Trig Functions
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Important Idea
These angles arespecial because theyhave exact value trig
functions.
8/7/2019 Lecture 1.41 Value of Trig Functions
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Consider the first two
special angles…
Long
S h o r
t
S i d e
H y p o t e n u s
e 6π
3π
8/7/2019 Lecture 1.41 Value of Trig Functions
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L
o n g S i d e
H y p o t e n
u s e 6
π
3
π
Short Side…
orientationdoes not
change therelationshi
psbetween
sides and
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 6/52
L o
n g S
i d e H y
p o t e n
u s e 6
π
3
π
Short Side
…
orientation does not
change therelationshi
ps betweensides and
angles
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 7/52
Analysis
Consideran
equilateral
trianglewithsides 2
2
2
2
8/7/2019 Lecture 1.41 Value of Trig Functions
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Analysis
...equilateral triangles
areequiangula
r and eachangle must
2
2
2
3
π
3π
3π
3π
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 9/52
Analysis
…theangle
bisectorintersects
the baseat themidpoint
22
1 1
3π
3π
6
π
6
π
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 10/52
Analysis
2 2
1 1
1
2?
?= 3
6
π
6
π
6
π
3π
3π
3
π
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 11/52
Try This
Find theheight of
anequilateral
trianglewith eachside 4
4 4
4
?
8/7/2019 Lecture 1.41 Value of Trig Functions
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Important Idea
In a rad triangle:, ,6 3 2
π π π
•the short side is one-half the hypotenuse.
•the long side istimes the short side.3 M
e m o r i z
e
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 13/52
Try This
Find thelength
of themissing
sides:
6
π
10
5
5 3
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 14/52
Try This
Find thelength
of themissing
sides:
3
π
5
5 3
3
10 3
3
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 15/52
Try This
What iswrong
with thispicture? 6
π
8/7/2019 Lecture 1.41 Value of Trig Functions
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Consider the last special
angle…
6
π
3
π
4
π
4π
4
π
8/7/2019 Lecture 1.41 Value of Trig Functions
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H y p o t e
n u s e
…
orientationdoes not
change therelationshi
psbetween
sides and
4
π
4
π
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 18/52
H y p o t e n
u s e
…
orientationdoes not
change therelationshi
psbetween
sides and
4
π
4
π
d
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 19/52
…andsidesoppositeequal
anglesare
equal... 4
π
4
π x
x
and by the
2 x
pythagorean theorem,
the hypotenuse is
8/7/2019 Lecture 1.41 Value of Trig Functions
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Important Idea
In a triangle:
• The legs of thetriangle are equal.
•the hypotenuse istimes the length of theleg.
M e m
o r i z
e2
, ,4 4 2
π π π
8/7/2019 Lecture 1.41 Value of Trig Functions
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Try This
Findthe
lengthof themissing
4
π
4
π
2
22 2
8/7/2019 Lecture 1.41 Value of Trig Functions
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Try This
Findthe
lengthof themissing
2
2
24
π
4
π
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
•
-1 1
-1
1
x
yr
θ
oppositesin
hypotenuse
y
r θ = =
adjacentcos
hypotenuse
x
r θ = =
oppositetan
adjacent
y
x
θ = =
•
• Memoriz
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
•
-1 1
-1
1
x
yr
θ
hypotenusecsc
opposite
r
yθ = =
hypotenusesec
adjacent
r
xθ = =
adjacentcot
opposite
x
y
θ = =
•
• Memoriz
8/7/2019 Lecture 1.41 Value of Trig Functions
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Find theexact
values of the sixtrig
3
π - 1 1
- 1
1Example
functions of thegiven special
angle in standard
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1
Analysis
1. Make
a righttriangle.
θ
rads3
π θ =
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 27/52
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1
Analysis
32
1
3. Use thedefinitions
of the trigfunctions to
find eachtrig function
exact value
θ
rads3
π θ =
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1
Analysis
32
1
opp 3sinhyp 2
θ = =
adj 1cos
hyp 2θ = =
opp 3tan 3
adj 1θ = = =
θ
rads3
π θ =
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1
Analysis
32
1
hyp 2 3cscopp 3
θ = =
hypsec 2
adjθ = =
adj 3cot
opp 3θ = =
θ
rads3
π θ =
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 31/52
Find theexact
values of the sixtrig
- 1 1
- 1
1Try This
functions of thegiven special
angle in standard
4π
8/7/2019 Lecture 1.41 Value of Trig Functions
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Solution
2sin4 2
π =
2cos
4 2
π =
tan 14
π =
csc 24
π =
sec 24
π =
cot 14
π =
8/7/2019 Lecture 1.41 Value of Trig Functions
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Important Idea
If the given angle isother than a quadrant Iangle, the exact valuesof the trig functions arecalculated using thereference angle of the
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
Reference Angle: theangle between a givenangle and the nearest x axis. (Note: x axis; not y
axis). Reference anglesare always positive.
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
How you find the
reference angledepends on whichquadrant contains thegiven angle.
R i
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
2
1
-1
-2
-2 -1 1 2 3
ReferenceAngle
GivenAngle &
R i
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
2
1
-1
-2
-3 -2 -1 1 2
Given
Angle
Reference
Angle
R i
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
-3 -2 -1 1 2
2
1
-1
-2
ReferenceAngle
Given
Angle
R i
8/7/2019 Lecture 1.41 Value of Trig Functions
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Review
-3 -2 -1 1 2
2
1
-1
-2
Reference
Angle
Given
Angle
E l Give
8/7/2019 Lecture 1.41 Value of Trig Functions
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ExampleDraw adiagramshowing the
relevantangles andsides andfind the sin,cos & tan of
56π
Given
Angle
Ref. Angle6
π
5
6
π
E l
8/7/2019 Lecture 1.41 Value of Trig Functions
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Example
5
6
π
6
π
Short
Side
LongSide
GivenAngl
e
E l
8/7/2019 Lecture 1.41 Value of Trig Functions
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Example
5
6
π
6
π
21
3−
Why is 1 & 2 pos &3
5 1sin
6 2
π =
5 3
cos 6 2
π = −
5 3
tan 6 3
π = −
T Thi Give
8/7/2019 Lecture 1.41 Value of Trig Functions
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Try ThisDraw adiagramshowing the
relevantangles andsides andfind the csc,sec & cot of
56π
Given
Angle
Ref. Angle6
π
5
6
π
S l ti
8/7/2019 Lecture 1.41 Value of Trig Functions
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Solution
5
6
π
6
π
21
3− 5csc 2
6
π =
5 2 3sec6 3π
= −
5cot 36
π = −
Find the Quadrant
al
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1Find the
values of the six trig
functionsof the
givenangle in
standard
2
π θ
sinθ
cosθ
tanθ
cscθ
secθ
cotθ
Quadrantal
Angles
Find the Quadranta
l Angle
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1Find the
values of the six trig
functionsof the
givenangle in
standard
2π
sinθ
cosθ
tanθ
cscθ
secθ
cotθ
Quadrantal Angle
Find the Quadranta
l Angle
8/7/2019 Lecture 1.41 Value of Trig Functions
http://slidepdf.com/reader/full/lecture-141-value-of-trig-functions 47/52
- 1 1
- 1
1Find the
values of the six trig
functionsof the
givenangle in
standard
π
sinθ
cosθ
tanθ
cscθ
secθ
cotθ
Quadrantal Angle
Find theQuadrantal A
ngles
8/7/2019 Lecture 1.41 Value of Trig Functions
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- 1 1
- 1
1Find the
values of the six trig
functionsof the
givenangle in
standard
32π
sinθ
cosθ
tanθ
cscθ
secθ
cotθ
Quadrantal Angles
Important Ideas
8/7/2019 Lecture 1.41 Value of Trig Functions
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Important Ideas
• Trig functions of quadrantal angles have
exact values.
• Trig functions of allother angles have
approximate values
• Trig functions of specialangles have exact values.
8/7/2019 Lecture 1.41 Value of Trig Functions
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Example
Use a calculator toapproximate cos to4 decimal places.
You did not forget tocheck “mode”, did
4
9
π
−
8/7/2019 Lecture 1.41 Value of Trig Functions
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Example
Use a calculator toapproximatecsc(11.55π ) to 4
decimal places.Check “mode” everytime.