Download 9.4/9.5 Warmup .

9.4/9.5 Warmup
Find the measure of the missing leg in the
𝒚𝟏
right triangle, and then calculate the ratio .
𝒙𝟏
1.
9
2.
The two triangles are _____________ so two
angles in each triangle are ___________.
March 28, 2016
Geometry 9.5 Trigonometric Ratios
1
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?
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Geometry 9.5 Trigonometric Ratios
3
Goals


Find the sine, cosine, and tangent of
an acute angle.
Solve problems using trigonometric
ratios.
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Geometry 9.5 Trigonometric Ratios
4
Terminology


No one, except stuffed-shirt
mathematics teachers, uses the
word trigonometry.
It’s
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Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
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Trig Ratios





Based on the sides of a right triangle.
We will study only three:
Sine
Cosine
Tangent
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Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
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Right Triangle
Opposite
A
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Adjacent
Geometry 9.5 Trigonometric Ratios
11
Trig Ratio Definition: Sine
Opposite
A
Adjacent
Opposite
Sine of A = Hypotenuse
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Geometry 9.5 Trigonometric Ratios
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Trig Ratio Definition: Cosine
Opposite
A
Adjacent
Adjacent
Cosine of A = Hypotenuse
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Geometry 9.5 Trigonometric Ratios
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Trig Ratio Definition: Tangent
Opposite
A
Adjacent
Opposite
Tangent of A = Adjacent
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Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
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Memory Aid



Sine is Opposite over Hypotenuse.
Cosine is Adjacent over Hypotenuse.
Tangent is Opposite over Adjacent.
SOH CAH TOA
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Geometry 9.5 Trigonometric Ratios
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Trig Ratios
A
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Geometry 9.5 Trigonometric Ratios
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Writing Ratios SOH CAH TOA
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
3?
sin A 
5?
4
?
cos A 
5?
A
3?
tan A 
4?
18
Writing Ratios SOH CAH TOA
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
3
sin A 
5
4
cos A 
5
A
3
tan A 
4
19
Example 1
Find sin S, cos S, and tan S. Write each
answer as a fraction and as a decimal
rounded to four places.
sin S =
80
40
=
= .9756
82
41
cos S =
18
82
tan S =
80
40
=
= 4.444
18
9
=
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9
41
= .2195
Geometry 9.5 Trigonometric Ratios
20
Your Turn
Find sin R, cos R, and tan R. Write each
answer as a fraction and as a decimal
rounded to four places.
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sin R =
18
82
cos R =
80
40
=
= .9756
82
41
tan R =
18
9
=
= .2250
80
40
Geometry 9.5 Trigonometric Ratios
=
9
41
= .2195
21
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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Geometry 9.5 Trigonometric Ratios
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Mode Setting





Press MODE
Use the cursor
arrows and move
to Degree.
Press ENTER.
Press 2nd Quit.
Press Clear
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Geometry 9.5 Trigonometric Ratios
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Using Trig Functions





To find the sin 78:
Press ‘sin’
Enter 78
Press ENTER.
Answer is .9781
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Geometry 9.5 Trigonometric Ratios
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Find these values:








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sin 15
cos 45
tan 45
cos 80
sin 10
tan 5
cos 60
sin 90








.2588
.7071
1
.1736
.1736
.0875
.5
1
Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
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Example 2
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 (SOH CAH TOA)
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Geometry 9.5 Trigonometric Ratios
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Example 2
Find x.
Write the equation and solve.
x
sin 28 
15
15sin 28  x
x
15
28
x  7.0
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Geometry 9.5 Trigonometric Ratios
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Example 3
Find y.
Write the equation and solve.
y
cos31 
56
56cos31  y
y  48.0
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56
31
y
Geometry 9.5 Trigonometric Ratios
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Example 4
Find a.
Write the equation and solve.
a
tan 40 
8
8 tan 40  a
a
8
a  6.7
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40
Geometry 9.5 Trigonometric Ratios
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Fraction Reminder
If
b
ac
8
2
4
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Then
b
ca
8
4
2
Geometry 9.5 Trigonometric Ratios
31
Example 5
Find a.
Write the equation and solve.
17
tan 40 
a
a tan 40  17
17
40
17
a
tan 40
x  20.3
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a
Geometry 9.5 Trigonometric Ratios
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Example 6
x
y
Find x & y.
x
tan 78 
150
150 tan 78  x
x  705.7
78
150
cos 78 
y
150
y
cos 78
y  721.5
150
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Geometry 9.5 Trigonometric Ratios
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Angle of Elevation
Angle of
Elevation
Horizontal
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Geometry 9.5 Trigonometric Ratios
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Angle of Depression
Horizontal
Angle of
Depression
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Example 7
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  8.0
h
15
30 yd
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Geometry 9.5 Trigonometric Ratios
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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?
42
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x ft
30 ft
Geometry 9.5 Trigonometric Ratios
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Solution
x
tan 42 
30
x  30 tan 42
 30(.9004)
 27
42
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x ft
30 ft
Geometry 9.5 Trigonometric Ratios
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Solution
x
tan 42 
30
x  30 tan 42
 30(.9004)
 27
42
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27 ft
30 ft
Geometry 9.5 Trigonometric Ratios
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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.
y
You ski about 3349 ft down the
mountain.
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Geometry 9.5 Trigonometric Ratios
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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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Geometry 9.5 Trigonometric Ratios
41
Homework
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Geometry 9.5 Trigonometric Ratios
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