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Trigonometry: The study of triangles (sides
and angles)
Trigonometry has been used for centuries in the study of:
astronomy
surveying
geography
engineering
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physics
B
opposite
A
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adjacent
C
B
adjacent
A
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opposite
C
B
opposite
A
opp
sin A 
hyp
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adjacent
adj
cos A 
hyp
C
opp
tan A 
adj
B
adjacent
A
opp
sin B 
hyp
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opposite
adj
cos B 
hyp
C
opp
tan B 
adj
B
opp
A
adj
C
SOH CAH TOA
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SOH
CAH
TOA
B
hy
p
10
A
8
op
p
C
6
adj
8
sin A 
10
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6
cos A 
10
8
tan A 
6
B
hyp
SOH
CAH
TOA
5
A
4
sin B 
5
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3
4
adj
C
opp
3
cos B 
5
4
tan B 
3
B
SOH
CAH
TOA
hy
p
5
adj
A
12
sin B 
13
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12
opp
5
cos B 
13
C
12
tan B 
5
Use a calculator to determine the following ratios.
Be sure your calculator is set to degrees.
sin 21° = 0.3584
cos 53° = 0.6018
tan 72° = 3.0777
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Determine the following angles (nearest degree).
sin A = 0.4142
A = sin-1(0.4142)
= 24°
cos B = 0.6820
B = cos-1(0.6820)
= 47°
tan C = 1.562
C = tan-1(1.562)
= 57°
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Determine the following angles (nearest degree).
7
sin A =
12
= 0.5833
4
cos B =
15
= 0.2666
15
tan C =
8
= 1.875
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A = sin-1(0.5833)
= 36°
B = cos-1(0.2666)
= 75°
C = tan-1(1.875)
= 62°
Example 1: Determine
side measure of A
8
tan A 
13
tan A = 0.6154
B
8 cm
opp
A
13 cm
A = tan-1(0.6154)
ad
j
A = 31.6°
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C
SOH
CAH
TOA
Example 2: Determine side b
SOH
CAH
TOA
b
tan B 
8
b
tan 55 
8
55º
8 cm
A
b = 8 tan 55°
b = 8 (1.428)
b = 11.4 cm
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B
ad
C j
b
opp
Example 3: Determine the measure of P.
12
cos P 
17
R
SOH
CAH
TOA
cos P = 0.70588
P = cos–1(0.70588)
Q
P = 45.1
P
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Ex. 4: In DPQR, Q = 90°.
a) Find sin R if PR = 8 cm and PQ = 4 cm.
R
4 1
sin R  
8 2
b) Find cos R .
RQ2 = 82 – 42
P
RQ2 = 64 – 16
6.93
cos R 
8
RQ2 = 48
cos R  0.87
RQ  48
RQ  6.9
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R  30
4 cm Q
Example 5: Determine the
measure of B
B
3 cm
5
tan B 
3
tan B = 1.6666
B =
tan-1(1.6666)
B = 59.0°
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A
5 cm
ad
C j
opp
SOH
CAH
TOA
1
Ex 6: The slope of a wheelchair ramps is
12
1
12
A
What angle does the
ramp make with the
ground?
1
tan A 
12
A = tan-1(0.0833)
tan A = 0.0833
A = 4.8°
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Ex. 7: From a distance of 20 m from the base a
lighthouse the angle of elevation to the top of a
lighthouse is 38º. Determine the height of the
lighthouse.
h
tan 38 
20
h = 20 tan 38°
h = 20 (0.7813)
hh
38º
20 m
h = 15.6
The lighthouse is 15.6 m high.
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