Righ
t
T
r
ia
n
gle T
r
ig
onom
e
try
—
F
u
ll De
pt
h S
tu
d
y
G
u
ide
(
NC
Ma
t
h
2
CDM
)
1
.
PYTHAGOREAN THEOREM
—
Dee
p
Di
v
e
Fi
n
di
n
g Mi
ss
i
n
g Side
s
F
or
a
ny
RIGHT
tr
ia
n
gle
,
w
i
t
h leg
s
a a
n
d b a
n
d h
ypot
e
nus
e c
(
t
he
s
ide
oppos
i
t
e
t
he
r
igh
t
a
n
gle
,
al
w
a
ys
t
he l
on
ge
st
s
ide
):
a
²
+
b
²
=
c
²
Fi
n
di
n
g
t
he h
ypot
e
nus
e
(
gi
v
e
n
b
ot
h leg
s
):
c
=
√
(
a
²
+
b
²
)
E
x
a
mp
le
:
leg
s
6
a
n
d
8
→
c
=
√
(
36
+
64
)
=
√
100
=
10
Fi
n
di
n
g a
m
i
ss
i
n
g leg
(
gi
v
e
n
h
ypot
e
nus
e
+
on
e leg
):
a
=
√
(
c
²
−
b
²
)
E
x
a
mp
le
:
h
ypot
e
nus
e
13
,
on
e leg
5
→
a
=
√
(
169
−
25
)
=
√
144
=
12
Ke
y
s
e
tup
r
e
m
i
n
de
r
:
al
w
a
ys
ide
nt
if
y
t
he h
ypot
e
nus
e FIRST
(
i
t
'
s
oppos
i
t
e
t
he
r
igh
t
a
n
gle
,
a
n
d i
t
'
s
i
so
la
t
ed b
y
i
ts
elf
on
on
e
s
ide
o
f
t
he e
qu
a
t
i
on
).
Pl
u
ggi
n
g a h
ypot
e
nus
e i
nto
t
he
wron
g
spot
i
s
t
he
most
c
ommon
e
rror
.
C
ommon
P
yt
hag
or
ea
n
tr
i
p
le
s
wort
h
m
e
mor
i
z
i
n
g
(
r
ec
o
g
n
i
z
i
n
g
t
he
s
e
s
a
v
e
s
t
i
m
e
):
3
-
4
-
5
(
a
n
d
mu
l
t
i
p
le
s
:
6
-
8
-
10
,
9
-
12
-
15
,
5
-
12
-
13
,
e
t
c
.)
5
-
12
-
13
8
-
15
-
17
7
-
24
-
25
C
onv
e
rs
e
o
f
t
he P
yt
hag
or
ea
n
The
or
e
m
The P
yt
hag
or
ea
n
The
or
e
m
t
ell
s
you
ab
out
s
ide
s
IF
you
al
r
ead
y
k
now
i
t
'
s
a
r
igh
t
tr
ia
n
gle
.
The CONVERSE le
ts
you
t
e
st
w
he
t
he
r
a
tr
ia
n
gle IS a
r
igh
t
tr
ia
n
gle
,
j
ust
f
rom
i
ts
t
h
r
ee
s
ide
le
n
g
t
h
s
.
R
u
le
:
Gi
v
e
n
t
h
r
ee
s
ide le
n
g
t
h
s
a
,
b
,
c
(
w
i
t
h c a
s
t
he l
on
ge
st
):
If a
²
+
b
²
=
c
²
→
t
he
tr
ia
n
gle IS a
r
igh
t
tr
ia
n
gle
If a
²
+
b
²
>
c
²
→
t
he
tr
ia
n
gle i
s
ACUTE
(
all a
n
gle
s
<
90
°)
If a
²
+
b
²
<
c
²
→
t
he
tr
ia
n
gle i
s
OBTUSE
(
on
e a
n
gle
>
90
°)
P
ro
ce
ss
:
1
.
Ide
nt
if
y
t
he l
on
ge
st
s
ide
—
call i
t
c
.
2
.
S
qu
a
r
e all
t
h
r
ee
s
ide
s
.
3
.
C
omp
a
r
e a
²
+
b
²
to
c
²
.
E
x
a
mp
le
:
s
ide
s
7
,
8
,
10
→
7²
+
8²
=
49
+
64
=
113
;
10²
=
100
.
Si
n
ce
113
>
100
,
t
hi
s
i
s
a
n
ACUTE
tr
ia
n
gle
(
not
r
igh
t
).
2
.
SPECIAL RIGHT TRIANGLES
—
Dee
p
Di
v
e
The
s
e a
r
e
r
igh
t
tr
ia
n
gle
s
w
h
os
e a
n
gle
m
ea
sur
e
s
c
r
ea
t
e
fix
ed
,
pr
edic
t
able SIDE RATIOS
—
m
ea
n
i
n
g
you
ca
n
fin
d all
s
ide
s
f
rom
j
ust
ONE k
nown
s
ide
,
w
i
t
h
out
n
eedi
n
g
tr
ig a
t
all
.
45
-
45
-
90
T
r
ia
n
gle
F
orm
ed b
y
c
utt
i
n
g a
squ
a
r
e i
n
half al
on
g i
ts
diag
on
al
.
T
wo
45
°
a
n
gle
s
(
i
sos
cele
s
)
a
n
d
on
e
90
°
a
n
gle
.
Side
r
a
t
i
o
:
leg
:
leg
:
h
ypot
e
nus
e
=
x
:
x
:
x
√
2
The
two
leg
s
a
r
e al
w
a
ys
EQUAL
(
s
i
n
ce
t
he
two
ac
ut
e a
n
gle
s
a
r
e e
qu
al
).
The h
ypot
e
nus
e
=
leg
×
√
2
Gi
v
e
n
a leg
,
fin
d
t
he h
ypot
e
nus
e
:
h
ypot
e
nus
e
=
leg
×
√
2
Gi
v
e
n
t
he h
ypot
e
nus
e
,
fin
d a leg
:
leg
=
h
ypot
e
nus
e
/
√
2
=
(
h
ypot
e
nus
e
×
√
2
)/
2
(
r
a
t
i
on
ali
z
ed
)
E
x
a
mp
le
:
leg
=
5
→
h
ypot
e
nus
e
=
5
√
2
E
x
a
mp
le
:
h
ypot
e
nus
e
=
10
→
leg
=
10
/
√
2
=
5
√
2
30
-
60
-
90
T
r
ia
n
gle
F
orm
ed b
y
c
utt
i
n
g a
n
e
qu
ila
t
e
r
al
tr
ia
n
gle i
n
half
.
A
n
gle
s
a
r
e
30
°,
60
°,
a
n
d
90
°.
Side
r
a
t
i
o
:
s
h
ort
leg
:
l
on
g leg
:
h
ypot
e
nus
e
=
x
:
x
√
3
:
2x
Sh
ort
leg
(
x
)
—
oppos
i
t
e
t
he
30
°
a
n
gle
L
on
g leg
(
x
√
3
)
—
oppos
i
t
e
t
he
60
°
a
n
gle
H
ypot
e
nus
e
(
2x
)
—
oppos
i
t
e
t
he
90
°
a
n
gle
,
al
w
a
ys
t
he l
on
ge
st
,
al
w
a
ys
d
ou
ble
t
he
s
h
ort
leg
Gi
v
e
n
t
he
s
h
ort
leg
:
l
on
g leg
=
s
h
ort
leg
×
√
3
,
h
ypot
e
nus
e
=
s
h
ort
leg
×
2
Gi
v
e
n
t
he
h
ypot
e
nus
e
:
s
h
ort
leg
=
h
ypot
e
nus
e
/
2
,
l
on
g leg
=
s
h
ort
leg
×
√
3
Gi
v
e
n
t
he l
on
g leg
:
s
h
ort
leg
=
l
on
g leg
/
√
3
(
r
a
t
i
on
ali
z
e
:
×
√
3
/
√
3
),
h
ypot
e
nus
e
=
s
h
ort
leg
×
2
E
x
a
mp
le
:
s
h
ort
leg
=
4
→
l
on
g leg
=
4
√
3
,
h
ypot
e
nus
e
=
8
Me
mory
t
i
p
:
i
n
30
-
60
-
90
,
t
he
s
ide
s
s
cale a
s
1
,
√
3
,
2
—
t
he h
ypot
e
nus
e i
s
al
w
a
ys
s
i
mp
le
st
to
spot
(
i
t
'
s
d
ou
ble
t
he
s
h
ort
e
st
s
ide
).
3
.
SOHCAHTOA
—
Dee
p
Di
v
e
F
or
a
r
igh
t
tr
ia
n
gle
,
tr
ig
r
a
t
i
os
r
ela
t
e a
n
ACUTE a
n
gle
to
t
he
r
a
t
i
o
o
f
two
o
f
t
he
tr
ia
n
gle
'
s
s
ide
s
.
Thi
s
on
l
y
wor
k
s
w
i
t
h a
r
igh
t
a
n
gle
pr
e
s
e
nt
.
Rela
t
i
v
e
to
a ch
os
e
n
ac
ut
e a
n
gle
θ
:
O
ppos
i
t
e
=
t
he
s
ide ac
ross
f
rom
θ
(
d
o
e
sn
'
t
tou
ch
t
he a
n
gle
)
Adjace
nt
=
t
he
s
ide
n
e
xt
to
θ
t
ha
t
i
s
NOT
t
he h
ypot
e
nus
e
H
ypot
e
nus
e
=
al
w
a
ys
t
he
s
ide
oppos
i
t
e
t
he
r
igh
t
a
n
gle
(
s
a
m
e
on
e e
v
e
ry
t
i
m
e
,
r
ega
r
dle
ss
o
f
w
hich ac
ut
e a
n
gle
you
p
ick
)
SOH
—
Si
n
e
s
i
n
(
θ
)
=
O
ppos
i
t
e
/
H
ypot
e
nus
e
CAH
—
C
os
i
n
e
c
os
(
θ
)
=
Adjace
nt
/
H
ypot
e
nus
e
TOA
—
Ta
n
ge
nt
t
a
n
(
θ
)
=
O
ppos
i
t
e
/
Adjace
nt
C
r
i
t
ical
st
e
p
bef
or
e ANY
tr
ig
pro
ble
m
:
r
elabel
t
he
s
ide
s
(
oppos
i
t
e
/
adjace
nt
/
h
ypot
e
nus
e
)
r
ela
t
i
v
e
to
w
hiche
v
e
r
sp
eci
fi
c a
n
gle
you
'
r
e
us
i
n
g
—
t
he
s
e label
s
CHANGE de
p
e
n
di
n
g
on
w
hich ac
ut
e a
n
gle i
s
t
he
r
efe
r
e
n
ce a
n
gle
.
The h
ypot
e
nus
e i
s
t
he
on
l
y
s
ide
w
h
os
e label
n
e
v
e
r
cha
n
ge
s
.
S
o
l
v
i
n
g f
or
a
m
i
ss
i
n
g
s
ide
:
1
.
Ide
nt
if
y
t
he k
nown
a
n
gle a
n
d
w
hich
two
s
ide
s
a
r
e i
nvo
l
v
ed
(
on
e k
nown
,
on
e
un
k
nown
).
2
.
Ch
oos
e
t
he c
orr
ec
t
r
a
t
i
o
(
SOH
,
CAH
,
or
TOA
)
ba
s
ed
on
w
hich
two
s
ide
s
you
ha
v
e
.
3
.
Se
t
up
t
he e
qu
a
t
i
on
a
n
d
so
l
v
e algeb
r
aicall
y
f
or
t
he
un
k
nown
s
ide
.
E
x
a
mp
le
:
a
n
gle
=
35
°,
h
ypot
e
nus
e
=
20
,
fin
d
t
he
s
ide
oppos
i
t
e
35
°.
s
i
n
(
35
°)
=
oppos
i
t
e
/
20
→
oppos
i
t
e
=
20
×
s
i
n
(
35
°)
≈
11
.
47
4
.
INVERSE TRIG
—
Dee
p
Di
v
e
Reg
u
la
r
tr
ig f
un
c
t
i
ons
(
s
i
n
,
c
os
,
t
a
n
)
t
ake a
n
ANGLE a
n
d gi
v
e
you
a
r
a
t
i
o
.
I
nv
e
rs
e
tr
ig
f
un
c
t
i
ons
d
o
t
he
oppos
i
t
e
—
t
he
y
t
ake a RATIO
(
f
rom
two
k
nown
s
ide
s
)
a
n
d gi
v
e
you
t
he
End of Document
333
