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9.4/9.5 Warmup
Find the measure of the missing leg in the
right triangle, and then calculate the ratio 𝒚𝟏
𝒙𝟏.
1. 2.
The two triangles are _____________ so two
angles in each triangle are ___________.
March 28, 2016 Geometry 9.5 Trigonometric Ratios 1
9
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Geometry
9.4 & 9.5 Trigonometric Ratios
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Essential Question
How is a right triangle used to find the sine,
cosine, and tangent of an acute angle?
March 28, 2016 Geometry 9.5 Trigonometric Ratios 3
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 4
Goals
Find the sine, cosine, and tangent of
an acute angle.
Solve problems using trigonometric
ratios.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 5
Terminology
No one, except stuffed-shirt
mathematics teachers, uses the
word trigonometry.
It’s
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 6
What is trig?
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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March 28, 2016 Geometry 9.5 Trigonometric Ratios 7
What you will learn…
The basic terms and methods of
solving right triangles.
How to use a calculator’s trig
functions.
How to solve problems using trig.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 8
Trig Ratios
Based on the sides of a right triangle.
We will study only three:
Sine
Cosine
Tangent
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 9
Right Triangle
Leg
Leg
A
From A, this leg is the Adjacent side.
From A, this leg is
the Opposite side.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 10
Right Triangle
Leg
Leg
A
From A, this leg is the Adjacent side.
From A, this leg is
the Opposite side.
B
From B, this leg is
the Adjacent side.
From B, this leg is the Opposite side.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 11
Right Triangle
Adjacent
Opposite
A
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 12
Trig Ratio Definition: Sine
Adjacent
Opposite
A
Sine of A =Opposite
Hypotenuse
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 13
Trig Ratio Definition: Cosine
Adjacent
Opposite
A
Cosine of A =Adjacent
Hypotenuse
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 14
Trig Ratio Definition: Tangent
Adjacent
Opposite
A
Tangent of A =OppositeAdjacent
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 15
Abbreviations
Tangent of A =OppositeAdjacent
Sine of A =Opposite
Hypotenuse
Cosine of A =Adjacent
Hypotenuse
sin A
cos A
tan A
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 16
Memory Aid
Sine is Opposite over Hypotenuse.
Cosine is Adjacent over Hypotenuse.
Tangent is Opposite over Adjacent.
SOH CAH TOA
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 17
Trig RatiosA
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 18
Writing Ratios SOH CAH TOA
4sin
5
3cos
5
4tan
3
B
B
B
3
4
5
A
B 3sin
5
4cos
5
3tan
4
A
A
A
?
?
?
?
?
?
?
?
?
?
?
?
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 19
Writing Ratios SOH CAH TOA
4sin
5
3cos
5
4tan
3
B
B
B
3
4
5
A
B 3sin
5
4cos
5
3tan
4
A
A
A
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Example 1
Find sin S, cos S, and tan S. Write each
answer as a fraction and as a decimal
rounded to four places.
March 28, 2016 Geometry 9.5 Trigonometric Ratios 20
sin S =
cos S =
tan S =
80
82=
40
41= .9756
18
82=
9
41= .2195
80
18=
40
9= 4.444
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Your Turn
Find sin R, cos R, and tan R. Write each
answer as a fraction and as a decimal
rounded to four places.
March 28, 2016 Geometry 9.5 Trigonometric Ratios 21
sin R =
cos R =
tan R =
80
82=
40
41= .9756
18
82=
9
41= .2195
18
80=
9
40= .2250
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 22
Calculators
Make sure your calculator is in
DEGREE mode.
Always use four decimal places of
accuracy when using trig functions.
All demonstrations here are from a TI
graphing calculator.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 23
Mode Setting
Press MODE
Use the cursor
arrows and move
to Degree.
Press ENTER.
Press 2nd Quit.
Press Clear
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 24
Using Trig Functions
To find the sin 78:
Press ‘sin’
Enter 78
Press ENTER.
Answer is .9781
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 25
Find these values:
sin 15
cos 45
tan 45
cos 80
sin 10
tan 5
cos 60
sin 90
.2588
.7071
1
.1736
.1736
.0875
.5
1
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 26
Solving Triangles
Carefully analyze the given
information.
Decide what you are trying to find.
Ask: Which trig function fits this
problem?
WRITE AN EQUATION. (SOH CAH TOA)
Solve.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 27
Example 2 Find x.
28
x 15
From the 28 angle, x is the ?
Opposite side,
and 15 is the
Hypotenuse.
What trig ratio is this?
Sine (SOH CAH TOA)
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 28
Example 2 Find x.
sin 2815
15sin 28
7.0
x
x
x
28
x 15
Write the equation and solve.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 29
Example 3 Find y.
cos3156
56cos31
48.0
y
y
y
31
y
56
Write the equation and solve.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 30
Example 4 Find a.
tan 408
8tan 40
6.7
a
a
a
40
a
8
Write the equation and solve.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 31
Fraction Reminder
bc a
82
4
ba c
84
2
If Then
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 32
Example 5 Find a.
17tan 40
tan 40 17
17
tan 40
20.3
a
a
a
x
40
a
17
Write the equation and solve.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 33
Example 6 Find x & y.
tan 78150
150 tan 78
705.7
x
x
x
78
x y
150
150cos78
150
cos78
721.5
y
y
y
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 34
Angle of Elevation
Horizontal
Angle of
Elevation
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 35
Angle of DepressionHorizontal
Angle of
Depression
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 36
Example 7
30 yd
15h
Standing 30 yards from a
tree, the angle of elevation
to the top of the tree is
15. How tall is the tree?
tan1530
30 tan15
8.0
h
h
h
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 37
Example 8
Isabella is 30 feet from a
fearsome monster. The angle
of elevation to the top of the
monster’s head is 42. How
tall is the monster?
30 ft42
x ft
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 38
Solution
30 ft42
x ft
tan 4230
30 tan 42
30(.9004)
27
x
x
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 39
Solution
30 ft42
27 ft
tan 4230
30 tan 42
30(.9004)
27
x
x
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Your Turn
You are skiing on a mountain. You start at an altitude
of 8400 feet and ski down to an altitude of 7200. The
angle of depression is 21°. Find the distance x you ski
down the mountain to the nearest foot.
March 28, 2016 Geometry 9.5 Trigonometric Ratios 40
y
You ski about 3349 ft down the
mountain.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 41
Summary
Trig ratios are based on acute angles
in right triangles.
They are Sine, Cosine, Tangent.
SOH CAH TOA
Angle of elevation is from the ground
up.
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March 28, 2016 Geometry 9.5 Trigonometric Ratios 42
Homework