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Geometry
9.5 Trigonometric Ratios
Goals
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I can find the sine, cosine, and tangent
of an acute angle.
I can solve problems using
trigonometric ratios.
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Terminology
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No one, uses the word
trigonometry.
It’s
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What is trig?
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Literally, the measure of triangles.
An extremely useful, practical and
powerful math tool.
A branch of math that finds its way into
practically everything we do.
Usually learned in high school.
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What you will learn…
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The basic terms and methods of
solving right triangles.
How to solve problems using trig.
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Trig Ratios
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Based on the sides of a right triangle.
We will study only three:
Sine
Cosine
Tangent
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Right Triangle
Leg
From A, this leg is
the Opposite side.
A
Leg
From A, this leg is the Adjacent side.
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Right Triangle
B
From B, this leg is
the Adjacent side.
Leg
From A, this leg is
the Opposite side.
A
Leg
From A, this leg is the Adjacent side.
From B, this leg is the Opposite side.
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Right Triangle
Opposite
A
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Adjacent
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Trig Ratio Definition: Sine
Opposite
A
Adjacent
Opposite
Sine of A = Hypotenuse
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Trig Ratio Definition: Cosine
Opposite
A
Adjacent
Adjacent
Cosine of A = Hypotenuse
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Trig Ratio Definition: Tangent
Opposite
A
Adjacent
Opposite
Tangent of A = Adjacent
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Abbreviations
Opposite
A = Hypotenuse
Sine sin
of A
Adjacent
cos
A
Cosine of A = Hypotenuse
Opposite
A = Adjacent
Tangent tan
of A
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Memory Aid
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Sine is Opposite over Hypotenuse.
Cosine is Adjacent over Hypotenuse.
Tangent is Opposite over Adjacent.
SOHCAHTOA
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Writing Ratios SOHCAHTOA
4?
sin B 
5?
?3
cos B 
5
?
4
?
tan B 
3
?
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B
5
3
4
Geometry 9.5 Trigonometric Ratios w/o Calculator
3?
sin A 
5?
4?
cos A 
5?
A
3?
tan A 
4?
15
Find these values using your
calculator—Make sure you’re in
Degree Mode!!!
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SIN 15
COS 45
TAN 45
COS 80
SIN 10
TAN 5
COS 60
SIN 90
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.2588
.7071
1
.1736
.1736
.0875
.5
1
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Solving Triangles
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Carefully analyze what given
information you have.
Decide what you are trying to find.
Ask: how do you use the given
information to solve the problem?
WRITE AN EQUATION. (SOHCAHTOA)
Solve.
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Example 1
Find x.
From the 28 angle, x is the ?
Opposite side,
and 15 is the
x
15
Hypotenuse.
What trig ratio is this?
28
Sine (SOHCAHTOA)
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Example 1
Find x.
Write the equation and solve.
x
sin 28 
15
15sin 28  x
15(.4695)  x
x
15
28
x  7.0
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Example 2
Find y.
Write the equation and solve.
y
cos31 
56
56cos31  y
56(.8572)  y
y  48.0
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56
31
y
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Example 3
Find a.
Write the equation and solve.
a
tan 40 
8
8 tan 40  a
8(.8391)  a
a  6.7
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a
40
8
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Algebra Reminder
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Solve:
12
3
x
12  3 x
a
a
c b
12 3 x

b
c
3
3
4x
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Example 4
Find a.
Write the equation and solve.
17
tan 40 
a
a tan 40  17
17
17
a

tan 40 .8391
x  20.3
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17
40
a
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Example 5
x
y
Find x & y.
150
x
cos
78

tan 78 
y
150
150 tan 78  x y  150  150
cos 78 .2079
150(4.7046)  x
y  721.5
x  705.7
78
150
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Angle of Elevation
Angle of
Elevation
Horizontal
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Angle of Depression
Horizontal
Angle of
Depression
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Problem
Standing 30 yards from a
tree, the angle of elevation
to the top of the tree is
15. How tall is the tree?
h
tan15 
30
h  30 tan15
h  30(.2679)
h  8.0
h
15
30 yd
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Summary
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Trig ratios are based on acute angles
in right triangles.
They are Sine, Cosine, Tangent.
SOHCAHTOA
Angle of elevation is from the ground
up.
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