Trigonometric Ratios

1
Trigonometric Ratios

• MM2G2. Students will define and apply sine,
  cosine, and tangent ratios to right triangles.
• MM2G2a Discover the relationship of
• the trigonometric ratios for
• similar triangles.

2
Trigonometric Ratios

• MM2G2b Explain the relationship between the
  trigonometric ratios of
• complementary angles.
• MM2G2c Solve application
• problems using the
• trigonometric ratios.

3
The following slides have been come from the
following sources

• www.mccd.edu/faculty/bruleym/.../trigonometric20r
  atios
• http//ux.brookdalecc.edu/fac/cos/lschmelz/Math20
  151/
• www.scarsdaleschools.k12.ny.us /202120915213753693
  /lib//trig.ppt
• Emily FreemanMcEachern High School

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Warm Up

• Put 4 30-60-90 triangles with the following sides
  listed and have students determine the missing
  lengths.

30 S 5 2 7v3 v2
90 H 10 4 14v3 2v2
60 L 5v3 2v3 21 v6
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Trigonometric Ratios

• Talk about adjacent and opposite sides have the
  kids line up on the wall and pass something from
  one to another adjacent and opposite in the room.
• Make a string triangle and talk about adjacent
  and opposite some more

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Trigonometric Ratios

• Determine the ratios of all the triangles on the
  board and realize there are only 3 (6?) different
  ratios.
• Talk about what it means for shapes to be
  similar.
• Make more similar right triangles on dot paper,
  measure the sides, and calculate the ratios.

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Trigonometric Ratios

• Try to have the students measure the angles of
  the triangles they made on dot paper.
• Do a Geosketch of all possible triangles and show
  the ratios are the same for similar triangles
• Finally name the ratios

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Warm Up

• Pick up a sheet of dot paper, a ruler, and
  protractor from the front desk.
• Draw two triangles, one with sides 3 4, and the
  other with sides 12 5
• Calculate the hypotenuse
• Calculate sine, cosine, and tangent for the acute
  angles.
• Measure the acute angles to the nearest degree.
• Show how to find sine, cosine, tangent of
  angles in the calculator

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Yesterday

• We learned the sine, cosine, and tangent of the
  same angle of similar triangles are the same
• Another way of saying this is The sine, cosine,
  tangent of congruent angles are the same

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Trigonometric Ratios in Right Triangles

• M. Bruley

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Trigonometric Ratios are based on the Concept of
Similar Triangles!
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All 45º- 45º- 90º Triangles are Similar!
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All 30º- 60º- 90º Triangles are Similar!
4
2
1
½
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All 30º- 60º- 90º Triangles are Similar!
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60º
2
60º
5
1
30º
30º
1
60º
30º
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In a right triangle, the shorter sides are called
legs and the longest side (which is the one
opposite the right angle) is called the hypotenuse
Well label them a, b, and c and the angles ? and
?. Trigonometric functions are defined by taking
the ratios of sides of a right triangle.
?
hypotenuse
c

First lets look at the three basic functions.
b
leg
SINE
COSINE
?
TANGENT
leg
a
They are abbreviated using their first 3 letters
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The Trigonometric Functions
SINE
COSINE
TANGENT
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SINE
Prounounced sign
18
COSINE
Prounounced co-sign
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TANGENT
Prounounced tan-gent
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Greek Letter q
Pronounced theta
Represents an unknown angle
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Greek Letter a
Pronounced alpha
Represents an unknown angle
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Greek Letter ß
Pronounced Beta
Represents an unknown angle
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hypotenuse
hypotenuse
opposite
opposite
adjacent
adjacent
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We could ask for the trig functions of the angle
? by using the definitions.
You MUST get them memorized. Here is a mnemonic
to help you.
?
c

The sacred Jedi word
b
SOHCAHTOA
SOHCAHTOA
?
INE
a
ANGENT
OSINE
PPOSITE
DJACENT
DJACENT
PPOSITE
YPOTENUSE
YPOTENUSE
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It is important to note WHICH angle you are
talking about when you find the value of the trig
function.
?
Let's try finding some trig functions with some
numbers. Remember that sides of a right triangle
follow the Pythagorean Theorem so
hypotenuse
c
5
b
4
opposite
?
a
3
Let's choose
sin ?
Use a mnemonic and figure out which sides of the
triangle you need for sine.
Use a mnemonic and figure out which sides of the
triangle you need for tangent.
tan ?
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You need to pay attention to which angle you want
the trig function of so you know which side is
opposite that angle and which side is adjacent to
it. The hypotenuse will always be the longest
side and will always be opposite the right angle.
This method only applies if you have a right
triangle and is only for the acute angles (angles
less than 90) in the triangle.
?
5
4
?
3
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We need a way to remember all of these ratios
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Question !
What is SohCahToa?
Is it in a tree, is it in a car, is it in the
sky or is it from the deep blue sea ?
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This is an example of a sentence using the word
SohCahToa.
I kicked a chair in the middle of the night and
my first thought was I need to SohCahToa.
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An example of an acronym for SohCahToa.
Seven old horses Crawled a hill To our
attic..
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Some
Old
Hippie
Came
A
Hoppin
Through
Our
Old Hippie
Apartment
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Sin
SOHCAHTOA
Opp
Hyp
Cos
Adj
Hyp
Tan
Opp
Adj
Old Hippie
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• Other ways to remember SOH CAH TOA
• Some Of Her Children Are Having Trouble Over
  Algebra.
• Some Out-Houses Can Actually Have Totally
  Odorless Aromas.
• She Offered Her Cat A Heaping Teaspoon Of Acid.
• Soaring Over Haiti, Courageous Amelia Hit The
  Ocean And ...
• Tom's Old Aunt Sat On Her Chair And Hollered. --
  (from Ann Azevedo)

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• Other ways to remember SOH CAH TOA
• Stamp Out Homework Carefully, As Having Teachers
  Omit Assignments.
• Some Old Horse Caught Another Horse Taking Oats
  Away.
• Some Old Hippie Caught Another Hippie Tripping On
  Apples.
• School! Oh How Can Anyone Have Trouble Over
  Academics.

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Trigonometry Ratios
Tangent A
Sine A
Cosine A
Soh Cah Toa
A
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14º
24º
60.5º
46º
82º
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The Tangent of an angle is the ratio of the
opposite side of a triangle to its adjacent side.
hypotenuse
1.9 cm
opposite
adjacent
14º
7.7 cm
0.25
Tangent 14º
0.25
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Tangent A
3.2 cm
24º
7.2 cm
0.45
Tangent 24º
0.45
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Tangent A
5.5 cm
46º
5.3 cm
1.04
Tangent 46º
1.04
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Tangent A
1.76
6.7 cm
Tangent 60.5º
1.76
60.5º
3.8 cm
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Tangent A
As an acute angle of a triangle approaches 90º,
its tangent becomes infinitely large
very large
Tan 89.9º 573
Tan 89.99º 5,730
etc.
very small
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Since the sine and cosine functions always have
the hypotenuse as the denominator, and since the
hypotenuse is the longest side, these two
functions will always be less than 1.
Sine A
Cosine A
Sine 89º .9998
A
Sine 89.9º .999998
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Sin a
7.9 cm
3.2 cm
24º
0.41
0.41
Sin 24º
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Cosine ß
7.9 cm
46º
5.5 cm
Cos 46º
0.70
0.70
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A plane takes off from an airport an an angle of
18º and a speed of 240 mph. Continuing at this
speed and angle, what is the altitude of the
plane after 1 minute?
After 60 sec., at 240 mph, the plane has
traveled 4 miles
4
x
18º
46
SohCahToa
Soh
Sine A
Sine 18
0.3090
1
x 1.236 miles or 6,526 feet
4
x
opposite
hypotenuse
18º
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An explorer is standing 14.3 miles from the base
of Mount Everest below its highest peak. His
angle of elevation to the peak is 21º. What is
the number of feet from the base of Mount
Everest to its peak?
Tan 21
0.3839
1
x 5.49 miles 29,000 feet
x
21º
14.3
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A swimmer sees the top of a lighthouse on the
edge of shore at an 18º angle. The lighthouse
is 150 feet high. What is the number of feet
from the swimmer to the shore?
0.3249x 150
Tan 18
0.3249
0.3249
0.3249
X 461.7 ft
1
150
18º
x
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A dragon sits atop a castle 60 feet high. An
archer stands 120 feet from the point on the
ground directly below the dragon. At what angle
does the archer need to aim his arrow to slay
the dragon?
Tan x
Tan x
0.5
Tan-1(0.5)
26.6º
60
x
120
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Solving a Problem withthe Tangent Ratio
We know the angle and the side adjacent to 60º.
We want to know the opposite side. Use
the tangent ratio
h ?
60º
53 ft
Why?
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Ex.
A surveyor is standing 50 feet from the base of a
large tree. The surveyor measures the angle of
elevation to the top of the tree as 71.5. How
tall is the tree?
tan 71.5
?
tan 71.5
71.5
y 50 (tan 71.5)
50
y 50 (2.98868)

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Ex. 5
A person is 200 yards from a river. Rather than
walk directly to the river, the person walks
along a straight path to the rivers edge at a
60 angle. How far must the person walk to reach
the rivers edge?
cos 60
x (cos 60) 200
200
x
60
x

X 400 yards
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Trigonometric Functions on a Rectangular
Coordinate System
Pick a point on the terminal ray and drop a
perpendicular to the x-axis.
r
y
x
The adjacent side is x The opposite side is y The
hypotenuse is labeled r This is called a
REFERENCE TRIANGLE.
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Trigonometric Ratios may be found by
Using ratios of special triangles
For angles other than 45º, 30º, 60º you will need
to use a calculator. (Set it in Degree Mode for
now.)