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13.1 – Use Trig with Right Triangles
Side opposite right angle
Opposite side:
Side opposite reference angle
Adjacent side:
Side next to reference angle,
not hypotenuse
Opposite
Hypotenuse:

Adjacent
adj
opp
cos  =
tan  =
hyp
adj
hyp
csc  =
opp
hyp
adj
sec  =
cot  =
adj
opp
Opposite
opp
sin  =
hyp

Adjacent
1
1
1
csc  =
sec  =
cot  =
sin 
cos 
tan 
.
csc = cosecant
sec = secant
cot = cotangent
.
.
SOH – CAH – TOA
1. Evaluate the six trigonometric functions of the angle .
O
H
12 A
c2 = a2 + b2
152 = a2 + 92
225 = a2 + 81
144 = a2
12 = a
9
sin  =
15
12
cos  =
15
9
tan  =
12
15
csc  =
9
15
sec  =
12
12
cot  =
9
2. Let  be an acute angle of a right triangle. Find the value of
the other five trigonometric functions of .
4
sin  
5
c2 = a2 + b2
52 = a2 + 4 2
O
H
5
4
25 = a2 + 16
9 = a2
3=a
4
sin  =
5
3
cos  =
5
4
tan  =
3
5
csc  =
4
5
sec  =
3
3
cot  =
4

3
A
2. Let  be an acute angle of a right triangle. Find the value of
the other five trigonometric functions of .
cot   3
adj
cot  =
opp
c2 = a2 + b2
c 1 
2
2
 3
2
c2 = 1 + 3
c2 = 4
c=2
4
sin  =
5
3
cos  =
5
4
tan  =
3
5
csc  =
4
5
sec  =
3
3
cot  =
4
O
1
H
2

3 A
3. Find the measure of the missing sides. Round to the nearest
hundredth.
O
H
A
SOH – CAH – TOA
c
d
c
sin 42° =
7
1
d
cos 42° =
7
1
7  sin 42° = c
7  cos 42° = d
4.68 = c
5.20 = d
3. Find the measure of the missing sides. Round to the nearest
hundredth.
H
A
O
a
b
tan 57° = 14
a
1
sin 57° = 14
b
1
a  tan 57° = 14
b  sin 57° = 14
a = 9.09
16.69 = b
SOH – CAH – TOA
4. Solve ABC, using the diagram and the given measurements.
A = 40°, c = 8
A
40°
H8
SOH – CAH – TOA
O
a
sin 40° = a
8
1
8  sin 40° = a
a = 5.14
b
cos 40° = b
8
1
8  cos 40° = b
6.13 = b
A
40°
B
50°
C
90°
a
5.14
b
6.13
c
8
Remember special triangles?
45°
45°
90°
30°
1
1
2
1
45°
60°
2
3
60°
1
2
2
1
45°
1
90°
30°
3
5. Find the exact values of the variables.
2
1
x = 13
y = 13 2
1
5. Find the exact values of the variables.
3
1
60°
2
x= 7 3
y = 14
5. Find the exact values of the variables.
3
x= 3
1
y= 3 3
2
5. Find the exact values of the variables.
3
1
2
18
3 18 3
x=
6 3


3
3
3
y = 12 3