Post on 21-Jan-2016
Right triangle trigonometry definitions.Build a right triangle in the diagram over the central angle.
A
B
C
a
b
cLabel the right triangle. a is the side OPPOSITE of .Label the right triangle using
theta( ) as the reference angle.
This angle is always ACUTE!
b is the side ADJACENT to .
c is the HYPOTENUSE.
sinThe length Opposite of .
The length of the Hypotenuse.
c
a
cosThe length Adjacent of .
The length of the Hypotenuse.
c
b
tanThe length Opposite of .
The length Adjacent of .
b
a
SOH-CAH-TOAI
N
E
P
P
O
S
I
T
E
Y
P
O
T
E
N
U
S
E
O
S
I
N
E
D
J
A
C
E
N
T
Y
P
O
T
E
N
U
S
E
A
N
G
E
N
T
P
P
O
S
I
T
E
D
J
A
C
E
N
T
Right triangle trigonometry definitions.Build a right triangle in the diagram over the central angle.
A
B
C
a
b
cLabel the right triangle.a is the side
Label the right triangle using
theta( ) as the reference angle.
This angle is always ACUTE!
b is the side
c is the HYPOTENUSE.
sinThe length of .
The length of the Hypotenuse.
c
a
cosThe length of .
The length of the Hypotenuse.
c
b
tanThe length of .
b
a
SOH-CAH-TOAI
N
E
P
P
O
S
I
T
E
Y
P
O
T
E
N
U
S
E
O
S
I
N
E
D
J
A
C
E
N
T
Y
P
O
T
E
N
U
S
E
A
N
G
E
N
T
P
P
O
S
I
T
E
D
J
A
C
E
N
T
OPPOSITE of .
OPPOSITEADJACENT of .
ADJACENT
Opposite
Adjacent
Opposite
The length of .Adjacent
c
b
c
a
a
b
REMEMBER
Pythagorean Theorem
222 cba
5
3sin
2
2
222
222
25
169
43
c
c
c
cba
5c
Find the 6 trigonometric functions with respect to .Circle the reference angle and label the opposite, adjacent, and hypotenuse.
OPP.
ADJ.
HYP.Find the value of the missing side, c. 5c
SOH-CAH-TOA
5
4cos
4
3tan
3
4cot
4
5sec
3
5csc
Special Right Triangle Relationships.
b
b
bb
45
45 c
22
222
2 cb
cbb
2
2 22
bc
bc
2b
90:45:452:: bbb
2:1:1
1
12
2
2
2
145sin
2
245cos
145tan 145cot
245sec
245csc
45
45
These answers are considered exact values.
Special Right Triangle Relationships.
a2
a2
a2
22
222
222
3
4
2
ab
aba
aba
3
3 22
ab
ab
90:60:30aaa 2:3:
2:3:1
60
60 60
b
a a60
30
3a
2
160
30
3
2
130sin
2
330cos
3
3
3
130tan 330cot
3
32
3
230sec
230csc
2
360sin
2
160cos
360tan 3
360cot
260sec
3
3260csc Do you see a pattern?
Cofunction Identities.
Two positive angles are complimentary if their sum is 90o. Our trigonometric functions are identified with the prefix “Co”.
Sine & Cosine
Tangent & Cotangent
Secant & Cosecant 90cossin
90sincos
90cottan
90tancot
90cscsec
90seccsc
From the 30-60-90 Ex.
2
130sin
2
160cos
3090cos
90sincos
5290sin52cos
38sin52cos
90cottan
7190cot71tan
19cot71tan
90cscsec 2490csc24sec
66csc24sec
90cossin
1590cos15sin 75cos15sin
202csc1090csc 2021090
20380
360 20
Definition of Reference Angle.
Let be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle formed by the terminal side of and the x – axis.
Find all six trigonometric function exact values of .210•Use special right triangles and reference angles to find the exact values of the trigonometric functions. 90:60:30
2:3:1 90:45:45
2:1:1SOH – CAH - TOA
S A
T C
301
3
2
opp
adj210cot
adj
opp210tan
31
3
90
180210
hyp
adj210cos
hyp
opp210sin
2
1
2
3
3
3
3
1
adj
hyp210sec
3
32
3
2
opp
hyp210csc 2
1
2
Reference angleopp
adj
hyp
Use special right triangles and reference angles to find the exact values of the trigonometric functions.
240cos 675tan
90:60:302:3:1
90:45:452:1:1
S A
T C
S A
T C
SOH – CAH - TOA
60
3
1
2
hyp
adj 240cos
2
1
451
1
2
tan 675opp
adj 1
1
1
240
Reference angleopp
adj
hyp
Reference angle
opp
adj
hyp
675
S A
T C
60
3
1
2120
Reference angleopp
adj
hyp S A
T C
603
1
2 opp
adj
hyp S A
T C
303
12
opp
adj
hyp
2
1 2
2
32
2
3
1
1 3 12
2 4 3
2
3
S A
T C
Give away for 45o angle
45135
135
45
225225
Right now we want DEGREE mode, move cursor to DEGREE and hit ENTER
2nd APPS activate the ANGLE window.Degree symbol.Minute symbol.
Second symbol.ALPHA, +
MODE
x-1 button
0.7571217563
977.97cos
1
-7.205879213
-0.4067366431
4283.51tan
1.253948151
96770915.0sin 1 4.75 '2475
1
0545829.1
cos
1
0545829.1
1cos "53'301851470432.18
sinWF 5.2sin2500 lb109
sinWF 5000sin 6.1 lb 531
right Cacute BA
complimentary
A
B
C
a
b
7.12c
5.34
_____
_____
____
b
a
B
SOH-CAH-TOA
opp.
adj.
hyp.
5.555.3490
5.55
Find angles first.
7.12
5.34sina
5.34sin7.12a in2.7
7.12
5.34cosb
5.34cos7.12b
in5.10
= 29.43 = 53.58
_____
_____
_____
b
B
A
Need to find the following:
A general rule is to always use the information you are given. We can find b by the Pythagorean Theorem.
22
222
acb
cba
22 43.2958.53 b
Sides were given to 2 decimal places … so b = 44.77
44.77
SOH-CAH-TOA
opp.
adj.
hyp.
58.53
43.29sin A
58.53
43.29sin 1A
Make sure the MODE is DEGREE.
3.33 7.563.33907.56
25
4020
h
opp.
adj.
hyp.
x
SOH-CAH-TOA
x
2520tan
xx
x 25
20tan
20tan
25
20tan
20tanx
20tan
25x
opp.
adj.
hyp.
= 68.687
687.68
40tany
y
y 40tan687.68
= 57.635
fth
h
635.82
635.5725
When a single angle is given, it is understood that the
Bearing is measured in a clockwise direction from due north.
S
45o
N
45o
N
165o
N
225o
Starts at a Bearing starting on the north-south line
and uses an acute angle to show the direction, either east or west. S 45o E
N
75o
N 75o W
BA
C
mi7.3
x29
miACx 2.3
61o
331o29
2961
61
opp.
hyp.
adj.SOH-CAH-TOA
7.3
29cosx
7.37.3
29cos7.3 x
S
P
C
117
c
47o
43o
43
473.5(22) = 77
4(22) = 88
222
222
8877 c
cps
22 8877 c nautical miles
SOH-CAH-TOA
opp.hyp.
adj. 5015tan
x
h
5050
15tan50
xx
hx
15tan50xh
Solve for x to find h.
opp.hyp.
adj.
x
h28tan x
x
hx 28tan
28tanxh
28tan15tan50 xx
28tan15tan5015tan xx 15tanx 15tanx
15tan28tan15tan50 xx
15tan28tan15tan50 x
15tan28tan 15tan28tan
15tan28tan
15tan50x
27h