Download Right Angle Trigonometry

Right Angle Trigonometry
Right Angle Trigonometry
What You Will Learn:
– To find values of the six trigonometric
functions for acute angles,
– To understand the two Special Trigonometric
triangles, and
– To solve problems involving right triangles.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
2
Right Angle Trigonometry
Definition: Trigonometry – is the study of the
relationships among the angles and sides of a right
triangle.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
3
Right Angle Trigonometry
Labeling a Triangle
Given : Angle B
A
Hypotenuse
A
B
Opposite
Adjacent Side
c
a
b
C
Opposite
Adjacent Side
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
4
Right Angle Trigonometry
What makes Trigonometry work?
Similar Right Triangles
What is required for two right triangles to be similar?
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
 ABC   ADE
Given a
Opposite SideDE
Opposite SideBC
rightHypotenuse
triangle AD = HypotenuseAB
AB
BC
CA
=
=
EA
AD DE
AB
BC
=
AD DE
DE BC
=
AD AB
B
D
Divide by AB and
multiply by DE
A
E
C
Opposite SideDE
Opposite SideBC
HypotenuseAD = HypotenuseAB
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
Opposite SideDE
Opposite SideBC
HypotenuseAD = HypotenuseAB
 ABC   ADE
AB
BC
CA
=
=
AD DE
EA
AB
CA
=
AD EA
EA
CA
=
AD AB
Adjacent SideEA
Adjacent SideCA
HypotenuseAD = HypotenuseAB
B
D
Divide by AB and
multiply by EA
A
E
C
Adjacent SideEA
Adjacent SideCA
HypotenuseAD = HypotenuseAB
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
Opposite SideDE
Opposite SideBC
HypotenuseAD = HypotenuseAB
 ABC   ADE
AB
BC
CA
=
=
AD DE
EA
BC
CA
=
DE EA
BC DE
=
CA EA
Adjacent SideEA
Adjacent SideCA
HypotenuseAD = HypotenuseAB
Opposite SideBC
Opposite SideDE
=
Adjacent
Divide
bySide
BCCAandAdjacent SideEA
B
D
multiply by EA
A
E
C
Opposite SideBC
Opposite SideDE
Adjacent SideCA = Adjacent SideEA
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
8
Right Angle Trigonometry
Opposite SideDE
Opposite SideBC
HypotenuseAD = HypotenuseAB
Adjacent SideEA
Adjacent SideCA
HypotenuseAD = HypotenuseAB
Opposite SideBC
Opposite SideDE
Adjacent SideCA = Adjacent SideEA
No matter the length of the sides of
the right triangle, these ratios remain
equal for a given acute angle. So,
what does this imply?
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
A
B
D
E
C
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Right Angle Trigonometry
sin A =
Let’s
call


 cos A =

 tan A =
Opposite SideDE
Opposite SideBC
HypotenuseAD = HypotenuseAB
Adjacent SideEA
Adjacent SideCA
HypotenuseAD = HypotenuseAB
Opposite SideBC
Opposite SideDE
Adjacent SideCA = Adjacent SideEA
So, for every right triangle with an
These are the three basic
acute angle A, the various ratios of
trigonometric functions.
the opposite side, adjacent side, and
the hypotenuse are the same, no matter
the length of the sides of the triangle,
as long as the angles are the same and
A
the triangles are similar.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
B
D
E
C
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Right Angle Trigonometry
=
Each pairsin
ofAequal
trigonometric functions
cos A =
are called co-functions
of
the acute angles of the
tan A =
right triangle.
sin B =
Opposite SideCA
HypotenuseAB
= cos A
cos B =
Adjacent SideBC
HypotenuseAB
= sin A
tan B =
Opposite SideCA
Adjacent SideBC
1
= cot A
tan A
Opposite SideBC
HypotenuseAB
Adjacent SideCA
HypotenuseAB
Opposite SideBC
Adjacent SideCA
A
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
B
C
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Right Angle Trigonometry
Definition of Six Basic Trig Functions
Given : Angle A
sin A =
Opposite Side
Hypotenuse
cos A =
Adjacent Side
Hypotenuse
tan A =
Opposite Side
Adjacent Side
csc A =
Hypotenuse
Opposite Side
sec A =
Hypotenuse
Adjacent Side
cot A =
Adjacent Side
Opposite Side
A
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
B
c
a
b
C
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Right Angle Trigonometry
A mnemonic use to help remember the first three basic
trigonometric functions is:
SOH-CAH-TOA
Sine Opp over Hyp Tangent Opp over Adj
over Hyp
The cosecant Cosine
(csc) is Adj
the inverse
of the sine.
The secant (sec) is the inverse of the cosine.
The cotangent (cot) is the inverse of the tangent.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
What do the graphs of these trigonometric functions
look like?
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
14
Right Angle Trigonometry
sin 
The y-axis scale is –1.5
to 1.5, but what is the
maximum/minimum
value of the sine
function?
The x-axis scale is –2 to 2. Note
that it completes a cycle every 2
radians.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
15
Right Angle Trigonometry
sin 
To find
non-baseline
divide the
baseline
the 
The
number
of radiansperiods
a trig function
requires
to by
complete
coefficient.
3. The baseline
non-baseline
period
one cycle isExample:
called thesin
function’s
period.
Theis
2/3
or every
120.
period
cosecant
baseline
occurs
whenThe
thebaseline
coefficient
for foris the
1. The
sine’s
function
is the
same. is 0 to 2.
baseline period
is 2.
Its domain
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
16
Right Angle Trigonometry
cos 
The y-axis scale is –1.5
to 1.5, but what is the
maximum/minimum
value of the cosine
function?
The x-axis scale is –2 to 2. Note
that it completes a cycle every 2
radians.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
17
Right Angle Trigonometry
cos 
The cosine’s baseline period is 2.
Cosine domain is 0 to 2.
The baseline period for the secant function is the same.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
18
Right Angle Trigonometry
tan 
The y-axis scale is –1.5
to 1.5, but what is the
maximum/minimum
value of the tangent
function?
The x-axis scale is –2 to 2. Note
that it completes a cycle every 
radians.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
19
Right Angle Trigonometry
tan 
The tangent’s baseline period is .
Tangent domain is –/2 to /2.
The baseline period for the cotangent function is the same.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
20
Right Angle Trigonometry
csc 
Note the scale change.
The y-axis scale is –5 to
5, but what is the
maximum/minimum
value of the cosecant
function?
The x-axis scale is –2 to 2. Note
that it completes a cycle every 2
radians.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
21
Right Angle Trigonometry
sec 
The y-axis scale is –10
to 10, but what is the
maximum/minimum
value of the secant
function?
The x-axis scale is –2 to 2. Note
that it completes a cycle every 2
radians.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
22
Right Angle Trigonometry
cot 
The y-axis scale is –10
to 10, but what is the
maximum/minimum
value of the cotangent
function?
The x-axis scale is –2 to 2. Note
that it completes a cycle every 
radians.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
23
Right Angle Trigonometry
sin  and csc 
The x-axis scale is –2 to 2. Note that they complete a
cycle every 2 radians (right half of graph).
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
24
Right Angle Trigonometry
cos  and sec 
The x-axis scale is –2 to 2. Note that they complete a
cycle every 2 radians (right half of graph).
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
tan  and cot 
The x-axis scale is –2 to 2. Note that they complete a
cycle every 2 radians, and they are shifted /2 radians
from each other.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
26
Right Angle Trigonometry
Special Triangles
Given: Equilateral Triangle
AC = AB/2
B
AB2 = AC2 + BC2
Let x = AB
BC = (3/2) x
BC2 = AB2 – AC2
BC2 = AB2 – (AB/2)2
BC2 = AB2 –
AB2/
BC2 = (3/4) AB2
BC = (3/2) AB
19 July 2011
4
A
C
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
D
27
Right Angle Trigonometry
Special Triangles
Given: Equilateral Triangle
B
Let x = AB
BC = (3/2) x
AC = x/2
2w
x
These relationships
Which
relationship
are true
for any 30you
use depends
on
60oproblem.
triangle
the
(3/2 ) x
A
D
C
x/
19 July 2011
w3
2
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
w
28
Right Angle Trigonometry
For example: given a 30-60 triangle with the hypotenuse
of length 10 units, what are the lengths of the other two
sides?
The largest side is
across from which
angle?
The 30-60 triangle
relationship used
was:
19 July 2011
30o
10
x
x3
53
/2
60o
10/ x = 5
2 /2
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
29
Right Angle Trigonometry
Suppose we only knew the short side and its length is 9.
What is the length of the other side and the hypotenuse?
30o
2  9 = 18
2x
The 30-60 triangle
relationship used
was:
93
x3
60o
x
9
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
30
Right Angle Trigonometry
Suppose we only knew the long side and its length is 7.
What is the length of the other side and the hypotenuse?
x3 = 7
x = 7/3
30o
2  7/3 = 14/3
7
60o
7/
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
3
31
Right Angle Trigonometry
Special Triangles
Given: Right Isosceles Triangle
AC = BC
AB2 = AC2 + BC2
B
AB2 = AC2 + AC2
AB2
=
x 2
2AC2
x
AB = AC2
Let x = AC = BC
A
Then AB = x 2
19 July 2011
C
x
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
32
Right Angle Trigonometry
Special Triangles
Given: Right Isosceles Triangle
Another Form
Given AB = x
B
x
x
2
A
C
x
2
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
33
Right Angle Trigonometry
3
7
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
B
34
Right Angle Trigonometry
3
7
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
B
35
Right Angle Trigonometry
3
7
B
The answer is D.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
36
Right Angle Trigonometry
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
37
Right Angle Trigonometry
What You Have Learned:
– To find values of trigonometric functions for
acute angles, and
– To solve problems involving right triangles.
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
38
Right Angle Trigonometry
END OF LINE
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
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19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
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Right Angle Trigonometry
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
41
Right Angle Trigonometry
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
42
Right Angle Trigonometry
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
43
Right Angle Trigonometry
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
44
Right Angle Trigonometry
19 July 2011
Alg2_13_01_RightAngleTrig.ppt
Copyrighted © by T. Darrel Westbrook
45