Download Trigonometric Functions

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.
Trigonometric Ratios
 MM2G2b:
Explain the relationship
between the trigonometric ratios of
complementary angles.
 MM2G2c: Solve application
problems using the
trigonometric ratios.
The following slides have been
come from the following sources:
www.mccd.edu/faculty/bruleym/.../trigo
nometric%20ratios
http://ux.brookdalecc.edu/fac/cos/lschm
elz/Math%20151/
www.scarsdaleschools.k12.ny.us
/202120915213753693/lib/…/trig.ppt
Emily Freeman
McEachern High School
Warm Up
Put 4 30-60-90 triangles with the following
sides listed and have students determine
the missing lengths.
30
S
5
2
7√3
√2
90
H
10
4
14√3
2√2
60
L
5√3
2√3
21
√6
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
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.
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
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
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
Trigonometric Ratios
in Right Triangles
M. Bruley
Trigonometric Ratios are
based on the Concept of
Similar Triangles!
All 45º- 45º- 90º Triangles are Similar!
2
45 º
1
1
45 º
1
2
1
2
1
2 2
45 º
2
2
All 30º- 60º- 90º Triangles are Similar!
2 30º
3
30º
60º
4
2 3
1
60º
1
30º
60º
½
2
3
2
All 30º- 60º- 90º Triangles are Similar!
2
60º
10
1
30º
30º
3
5 3
1
60º
30º
3
2
1
2
60º
5
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
We’ll label them a, b, and c and the angles 
and . Trigonometric functions are defined by
taking the ratios of sides of a right triangle.

adjacent
c
First let’s look at the three basic functions.
leg
b
SINE

leg a
COSINE
TANGENT
They are abbreviated using their first 3 letters
opposite
a
opposite a
sin  

tan  

hypotenuse c
adjacent b
adjacent
b
cos  

hypotenuse c
The Trigonometric Functions
SINE
COSINE
TANGENT
SINE
Prounounced
“sign”
COSINE
Prounounced
“co-sign”
TANGENT
Prounounced
“tan-gent”
Greek Letter q
Pronounced
“theta”
Represents an unknown angle
Greek Letter α
Pronounced
“alpha”
Represents an unknown angle
Greek Letter β
Pronounced
“Beta”
Represents an unknown angle
Opp
Sin 
Hyp
Adj
Cos 
Hyp
Opp
Tan 
Adj
hypotenuse
hypotenuse
q
adjacent
opposite
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

adjacent
a
opposite
b
sin  

hypotenuse c
adjacent
a
cos  

hypotenuse c
opposite b
tan  

adjacent a
It is important to note WHICH angle you are talking
about when you find the value of the trig function.

c
5
4b
Let's try finding some trig functions
with some numbers. Remember that
sides of a right triangle follow the
Pythagorean Theorem so
a b c
2

2
2
adjacent
a3
sin  =
o 3

h 5
Let's choose:
tan  =
o 4

a 3
32  4 2  5 2
Use a mnemonic and
figure out which sides
of the triangle you
need for tangent.
sine.
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.

Oh,
I'm
acute!
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
So
am I!
We need a way
to remember
all of these
ratios…
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 ?
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.
An example of an acronym for SohCahToa.
Seven
old
horses
Crawled
a
hill
To
our
attic..
Some
Old
Hippie
Came
A
Hoppin’
Through
Our
Old Hippie Apartment
SOHCAHTOA
Old Hippie
Sin
Opp
Hyp
Cos
Adj
Hyp
Tan
Opp
Adj
Other ways to remember SOH CAH TOA
1. Some Of Her Children Are Having Trouble
Over Algebra.
2. Some Out-Houses Can Actually Have
Totally Odorless Aromas.
3. She Offered Her Cat A Heaping Teaspoon
Of Acid.
4. Soaring Over Haiti, Courageous Amelia Hit
The Ocean And ...
5. Tom's Old Aunt Sat On Her Chair And
Hollered. -- (from Ann Azevedo)
Other ways to remember SOH CAH TOA
1. Stamp Out Homework Carefully, As Having
Teachers Omit Assignments.
2. Some Old Horse Caught Another Horse
Taking Oats Away.
3. Some Old Hippie Caught Another Hippie
Tripping On Apples.
4. School! Oh How Can Anyone Have Trouble
Over Academics.
Trigonometry Ratios
opposite
Tangent A =
adjacent
opposite
Sine A =
hypotenuse

adjacent
Cosine A =
hypotenuse

Soh Cah Toa
A
14º
24º
60.5º
46º
82º
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
1.9
7.7

0.25
Tangent 14º
 0.25
opposite
Tangent A =
adjacent
3.2 cm

7.2 cm
3.2
7.2

0.45
24º
Tangent 24º
 0.45
opposite
Tangent A =
adjacent
5.5 cm

46º
5.5
5.3

5.3 cm
1.04
Tangent 46º
 1.04
opposite
Tangent A =
adjacent
6.7 cm

6.7
3.8

1.76
Tangent 60.5º  1.76


60.5º
3.8 cm
opposite
Tangent A =
adjacent
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
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.
opposite
Sine A =
hypotenuse
adjacent
Cosine A =
hypotenuse

Sine 89º = .9998
A
Sine 89.9º = .999998
Sin α =
opposite
hypotenuse
7.9 cm
3.2 cm
24º
3 .2
7 .9

0.41
Sin 24º
 0.41
adjacent
Cosine β =
hypotenuse
7.9 cm
46º
5 .5
7 .9

5.5 cm
0.70
Cos 46º
 0.70
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
x
4
18º
SohCahToa
Soh
Sine A =
opposite
hypotenuse

x
opposite
x
Sine 18 =
4
x
0.3090 =
4
1
x = 1.236 miles

or
4
6,526 feet
hypotenuse

18º
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?
x
x
Tan 21 =
0.3839 =
14.3 1
14.3
x
x = 5.49 miles
= 29,000 feet


14.3
21º
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?
150
150
Tan 18 =
x
0.3249 =150
x
1


x
0.3249x = 150
0.3249 0.3249
X = 461.7 ft
18º
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?
60
Tan x =
120
Tan x = 0.5
Tan-1(0.5) = 26.6º

60
x
120
Solving a Problem with
the Tangent Ratio
We know the angle and the
side adjacent to 60º. We want to
know the opposite side. Use the
tangent ratio:
h=?
2
3
60º
53 ft
1
tan 60 
opp h

adj 53
3 h
Why?

1 53
h  53 3  92 ft
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°
?
50
71.5
°
Opp

Hyp
y

50
tan
71.5°
y = 50 (tan 71.5°)
y = 50 (2.98868)
y  149.4 ft
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 river’s edge at a 60° angle.
How far must the person walk to reach the river’s
edge?
cos 60°
x (cos 60°) = 200
200
60°
x
x
X = 400 yards
Trigonometric Functions on a
Rectangular Coordinate System
y
Pick a point on the
terminal ray and drop a
perpendicular to the x-axis.
r
y
q
x
x
The adjacent side is x
The opposite side is y
The hypotenuse is labeled r
This is called a
REFERENCE TRIANGLE.
y
r
x
cos q 
r
y
tan q 
x
sin q 
r
y
r
sec q 
x
x
cot q 
y
csc q 
Trigonometric Ratios may be found by:
Using ratios of special triangles
2
45 º
1
1
1
2
1
cos 45 
2
tan 45  1
sin 45 
For angles other than 45º, 30º, 60º you will need to use a
calculator. (Set it in Degree Mode for now.)