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Right Triangle Trigonometry
Section 13.1
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13.1 Right Triangle Trigonometry
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Definitions
• Trigonometry
– Comes from Greek word – Trigonon, which
means 3 angles
– “Metry” means measure in Greek
• Trigonometry Ratios
– Sine, Cosine, Tangent, Secant, Cosecant,
Cotangent
• Types of angles
– Acute: Less than 90°
– Equilateral: 90°
– Obtuse: More than 90° but less than 180°
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13.1 Right Triangle Trigonometry
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Right Triangles
hypotenuse
opposite

adjacent
• Consider a right triangle, one of whose acute angles
is ө
• The three sides of a triangle are hypotenuse,
opposite, and adjacent side of ө
• To determine what is the opposite side, look at ө and
extend the line to determine the opposite
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13.1 Right Triangle Trigonometry
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Right Triangles
SOHCAHTOA
opposite
Sine ө=
hypotenuse
SIN
Cosine ө =
COS
adjacent
opposite
Tangent ө =
adjacent
hypotenuse
TAN
Reciprocals of SOHCAHTOA
adjacent
hypotenuse
hypotenuse
Cosecant ө =
Secant ө =
Cotangent ө=
adjacent
opposite
opposite
CSC
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SEC
13.1 Right Triangle Trigonometry
COT
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Right Triangles
Relationships of Trigonometric Ratios
Sine ө =
SIN
opposite
hypotenuse
Cosine ө =
COS
Cosecant ө =
CSC
adjacent
hypotenuse
Tangent ө =
TAN
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opposite
adjacent
Secant ө =
SEC
hypotenuse
adjacent
Cotangent ө =
COT
13.1 Right Triangle Trigonometry
hypotenuse
opposite
adjacent
opposite
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Steps in Determining Triangles
1. Solve for x, using Pythagorean
Theorem
2. Determine the hypotenuse and the
opposite by identifying ө
3. Use Trigonometry Functions to
find what’s needed
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13.1 Right Triangle Trigonometry
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Example 1
Find x and
determine all
trig functions of
ө
13

Step 1: Find x
12
x
Use the Pythagorean Theorem to find the length of the adjacent side…
a2 + 122 = 132
a2 = 25
a=5
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13.1 Right Triangle Trigonometry
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Example 1
Find x and
determine all
trig functions of
ө
13

12
x
Step 2: Determine the hypotenuse
and the opposite by identifying ө
adj = 5
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opp = 12
13.1 Right Triangle Trigonometry
hyp = 13
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Example 1
Find x and determine all trig
functions of ө
13

12
5
Step 3: Use Trigonometry Functions to find what’s needed
Sine ө=
SIN
opposite
hypotenuse
Cosine ө =
COS
adjacent Tangent ө = opposite
adjacent
hypotenuse
TAN
Cosecant ө =hypotenuse Secant ө =hypotenuse Cotangent ө=
CSC
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opposite
SEC
adjacent
13.1 Right Triangle Trigonometry
COT
adjacent
opposite
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Example 1
Find x and determine all trig
functions of ө
13

12
5
Step 3: Use Trigonometry Functions to find what’s needed
Sine ө=
SIN
12
13
Cosecant ө =
CSC
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Cosine ө =
COS
13
12
Secant ө =
SEC
5
13
13
5
13.1 Right Triangle Trigonometry
Tangent ө =
TAN
Cotangent ө=
COT
12
5
5
12
10
Your Turn
Determine all trig
functions of ө
2

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3
1
13.1 Right Triangle Trigonometry
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Your Turn
Determine all trig
functions of ө
2

Sine ө=
SIN
Cosecant ө =
CSC
3
2
3
1
Cosine ө =
COS
2
3
Secant ө =
SEC
1
2
2
Tangent ө =
3
TAN
Cotangent ө=
COT
1
3
Can we have radicals in the denominators?
Actually, with trig ratios, it is accepted in the subject area. But it is
necessary to simplify radicals
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13.1 Right Triangle Trigonometry
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Example 2
What is given?
– Hypotenuse: 74
– Opposite of 30°: x
– Adjacent: Unknown
Solve for x.
Which of the six trig
ratios is best fit for
this triangle? (there
can be more than one
answer)
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13.1 Right Triangle Trigonometry
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Example 2
Which of the six trig ratios is
best fit for this triangle?
(there can be more than one
answer)
Solve for x.
opposite
sin  
hypotenuse
opposite
sin 30 
hypotenuse
x
sin 30 
Must change
74
the answer to
DEGREE
x  74sin 30
mode and not
RADIAN
x  37
mode in
calculator
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13.1 Right Triangle Trigonometry
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Example a
Find the value of sine, cosine and tangent
functions
Example 3
In a waterskiing competition, a jump ramp has
the measurements shown. To the nearest
foot, what is the height h above water
that a skier leaves the ramp?
opposite
Substitute 15.1° for θ, h for opposite,
hypotenuse
and 19 for hypotenuse.
h
sin15.1 
Multiply both sides by 19.
19
Use a calculator to simplify.
h  4.9496
sin  
The height above the water is about 5 ft.
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13.1 Right Triangle Trigonometry
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Your Turn
Solve for h. Round to 4 decimal
places
0.6765 km
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13.1 Right Triangle Trigonometry
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Your Turn
Solve for the rest of missing sides
of triangle ABC, given that A = 35°
and c = 15.67. Round to 4 decimal
places
B  55
a  8.9879
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b  12.8361
13.1 Right Triangle Trigonometry
C  90
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Example b
Find the value of x
10
Example c
Find the value of x.
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Example d
Find the value of x.
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Angle of Elevation vs. Depression
• Angle of Elevation is a measurement above
the horizontal line
• Angle of Depression is a measurement below
the horizontal line
Angle of Elevation
Angle of Depression
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6.2 Trig Applications
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Example 4
A flagpole casts a 60-foot shadow when the
angle of elevation of the sun is 35°. Find the
height of the flagpole.
35°
---- 60 Feet ---5/24/2017 1:24 PM
6.2 Trig Applications
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Example 4
A flagpole casts a 60-foot shadow when the
angle of elevation of the sun is 35°. Find the
height of the flagpole.
x
tan 35 
60
35°
---- 60 Feet ----
x  42.0125 ft.
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6.2 Trig Applications
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Example 5
Find the distance of a boat from a lighthouse if
the lighthouse is 100 meters tall, and the
angle of depression is 6°.
6
100 ft.
6
?
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 951.4364 ft.
6.2 Trig Applications
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Your Turn
A man who is 2 m tall stands on horizontal
ground 30 m from a tree. The angle of
elevation of the top of the tree from his eyes
is 28˚. Estimate the height of the tree.
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 17.9513 ft.
6.2 Trig Applications
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Example e
Solve
45 ft
Assignment
Pg 933 3-25 odd
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13.1 Right Triangle Trigonometry
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