Download trigonometry_tan_intro

© T Madas
Enlargement
Scale Factor
In Proportion
Constant Ratio
8
4
3
6
© T Madas
Enlargement
Scale Factor
In Proportion
Constant Ratio
8
4
3
6
© T Madas
© T Madas
Hypotenuse
Lies opposite the right angle
The longest side of a right angled
triangle
© T Madas
“thita” is a Greek letter
we use to mark angles
Opposite
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
θ
Adjacent
© T Madas
“thita” is a Greek letter
we use to mark angles
θ
Adjacent
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
Opposite
© T Madas
“thita” is a Greek letter
we use to mark angles
Opposite
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
θ
Adjacent
© T Madas
“thita” is a Greek letter
we use to mark angles
θ
Adjacent
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
Opposite
© T Madas
“thita” is a Greek letter
we use to mark angles
Opposite
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
θ
Adjacent
© T Madas
“thita” is a Greek letter
we use to mark angles
θ
Adjacent
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
Opposite
© T Madas
“thita” is a Greek letter
we use to mark angles
Opposite
Opposite side:
it lies opposite the angle θ
Adjacent side:
it touches the angle θ
The hypotenuse is always
the same but the other 2
sides change if θ changes
θ
Adjacent
This will become
important later
© T Madas
The
Beginning
of
Trigonometry
© T Madas
Opposite
=
2
3
Opposite = 8
Adjacent
=
8
12
θ
Adjacent = 12
© T Madas
Opposite = 4
Opposite
Adjacent
=
4
6
=
2
3
Why is this angle also θ ?
θ
Adjacent = 6
θ
© T Madas
Opposite
Opposite = 6
Adjacent
=
6
9
=
2
3
θ
Adjacent = 9
θ
© T Madas
Opposite = 5
Opposite
Adjacent
=
2
5
=
3
7.5
θ
Adjacent = 7.5
θ
© T Madas
Opposite
Opposite
Adjacent
θ
θ
=
2
3
θ
θ
Adjacent
© T Madas
For a given acute angle of a right
angled triangle:
Opposite
Opposite
Adjacent
θ
= constant
tangent
θ
of θ
θ
θ
Adjacent
© T Madas
For a given acute angle of a right
angled triangle:
Opposite
Opposite
Adjacent
θ
=
θ
tangent
of θ
θ
θ
θ
Adjacent
© T Madas
Tangent
Practice
© T Madas
What is the tangent of θ ?
What is the tangent of a ?
3
a
5
4
θ
Opposite
tanθ =
Adjacent
= 3 = 0.75
4
Opposite
tana =
Adjacent
= 4 ≈ 1.33
3
© T Madas
What is the tangent of x ?
5
x
13
12
tanx =
Opp
= 12 = 2.4
Adj
5
© T Madas
What is the tangent of y ?
Opp
tany =
= 15 = 1.875
Adj
8
17
15
y
8
© T Madas
What is the tangent of θ ?
25
θ
7
24
tanθ =
Opp
= 24 ≈ 2.43
Adj
7
© T Madas
In every right angled triangle:
Opposite side
Adjacent side
= constant
Opposite side
Adjacent side
= tanθ
Every acute angle θ has its
tanθ (constant ratio) stored in
your calculator
© T Madas
SHIFT
ALPHA
x!
nPr
^
nCr
a b/c
(–)
Rec(
x
c
A
:
Pol(
REPLAY
^
d/
^
^
x-1
MODE CLR
x2
B
. , ,,
RCL ENG
^
log
sin
cos
3
x3
ex e
ln
(
)
;
X
,
Y
M- M
M+
OFF
8
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
0
.
Out
NOW!
tan
7
Rnd
Calculators
C sin-1 D cos-1 E tan-1 F
hyp
STO
10x
ON
EXP
Ans
AC
=
© T Madas
SHIFT
ALPHA
x!
nPr
^
nCr
a b/c
(–)
Rec(
x
c
A
:
Pol(
REPLAY
^
d/
^
^
x-1
MODE CLR
x2
B
. , ,,
RCL ENG
^
log
sin
cos
3
x3
ex e
ln
C sin-1 D cos-1 E tan-1 F
hyp
STO
10x
ON
(
)
;
X
,
Y
tan
M- M
M+
OFF
7
8
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
Rnd
0
.
EXP
Ans
Find the tangent
button in your
calculator
AC
=
© T Madas
ta 30
0.5773502
n
69
SHIFT
ALPHA
x!
nPr
^
^
nCr
a b/c
(–)
Rec(
x
c
A
:
Pol(
REPLAY
^
d/
MODE CLR
^
x-1
x2
B
. , ,,
RCL ENG
10x
^
log
sin
cos
x3
ex e
ln
(
)
;
X
,
Y
tan
M- M
M+
OFF
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
.
=
3
8
0
0
ON
7
Rnd
3
C sin-1 D cos-1 E tan-1 F
hyp
STO
tan
EXP
Ans
AC
=
© T Madas
ta 64
2.0503038
n
42
SHIFT
ALPHA
x!
nPr
^
^
nCr
a b/c
(–)
Rec(
x
c
A
:
Pol(
REPLAY
^
d/
MODE CLR
^
x-1
x2
B
. , ,,
RCL ENG
10x
^
log
sin
cos
x3
ex e
ln
(
)
;
X
,
Y
tan
M- M
M+
OFF
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
.
=
3
8
0
4
ON
7
Rnd
6
C sin-1 D cos-1 E tan-1 F
hyp
STO
tan
EXP
Ans
AC
=
© T Madas
ta 29
0.5543090
n
51
SHIFT
ALPHA
x!
nPr
^
^
nCr
a b/c
(–)
Rec(
x
c
A
:
Pol(
REPLAY
^
d/
MODE CLR
^
x-1
x2
B
. , ,,
RCL ENG
10x
^
log
sin
cos
x3
ex e
ln
(
)
;
X
,
Y
tan
M- M
M+
OFF
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
.
=
3
8
0
9
ON
7
Rnd
2
C sin-1 D cos-1 E tan-1 F
hyp
STO
tan
EXP
Ans
AC
=
© T Madas
tan 0
5
.
-1
26.565051
18
SHIFT
ALPHA
x!
nPr
^
^
nCr
a b/c
(–)
Rec(
x
c
A
:
Pol(
REPLAY
^
d/
MODE CLR
^
x-1
x2
B
. , ,,
RCL ENG
10x
^
log
sin
cos
(
)
;
X
,
Y
x3
ex e
ln
tan
M- M
M+
OFF
8
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
0
.
EXP
Ans
0
.
5
=
3
7
Rnd
tan
ON
C sin-1 D cos-1 E tan-1 F
hyp
STO
shift
AC
You can use the calculator
to work backwards from a
tangent to an angle
This is known as the inverse
of the tangent
It is written as :
tan-1
Which acute angle in a right
angled triangle has tangent
equal to ½ ?
=
© T Madas
tan 4.
7
01
-1
76.315522
16
SHIFT
ALPHA
x!
nPr
^
nCr
x
c
a b/c
A
(–)
Rec(
^
d/
:
Pol(
REPLAY
^
x-1
MODE CLR
^
x2
B
. , ,,
STO
RCL ENG
10x
^
log
sin
cos
ON
3
x3
ex e
ln
C sin-1 D cos-1 E tan-1 F
hyp
(
)
;
X
,
Y
tan
M- M
M+
OFF
7
8
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
Rnd
0
.
EXP
Ans
tan
shift
AC
4
.
0
1
7
=
You can use the calculator
to work backwards from a
tangent to an angle
This is known as the inverse
of the tangent
It is written as :
tan-1
Which acute angle in a right
angled triangle has tangent
equal to 4.017 ?
=
© T Madas
tan 0
533
.
-1
28.057615
73
SHIFT
ALPHA
x!
nPr
^
nCr
x
c
a b/c
A
(–)
Rec(
^
d/
:
Pol(
REPLAY
^
x-1
MODE CLR
^
x2
B
. , ,,
RCL ENG
^
log
sin
cos
ON
3
x3
ex e
ln
C sin-1 D cos-1 E tan-1 F
hyp
STO
10x
(
)
;
X
,
Y
tan
M- M
M+
OFF
7
8
9
DEL
4
5
6
x
÷
1
2
3
+
–
Ran#
π
DRG›
%
Rnd
0
.
EXP
Ans
tan
shift
AC
0
.
5
3
3
=
You can use the calculator
to work backwards from a
tangent to an angle
This is known as the inverse
of the tangent
It is written as :
tan-1
Which acute angle in a right
angled triangle has tangent
equal to 0.533 ?
=
© T Madas
© T Madas