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.
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Trigonometric RatiosMM2G2. 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 RatiosMM2G2b: 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/.../trigonometric%20ratios
http://ux.brookdalecc.edu/fac/cos/lschmelz/Math%20151/
www.scarsdaleschools.k12.ny.us /202120915213753693/lib/…/trig.ppt
Emily FreemanMcEachern High School
Warm UpPut 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 RatiosTalk 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 RatiosDetermine 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 RatiosTry 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 UpPick 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 & 5Calculate the hypotenuseCalculate 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
YesterdayWe 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!
45 º
2
2
22
45 º
1
1
2
45 º
1
2
1
2
1
All 30º- 60º- 90º Triangles are Similar!
1
60º
30º
½
23
32
60º
30º
2
4
2
60º
30º
1
3
All 30º- 60º- 90º Triangles are Similar!
10 60º
30º
5
35
2 60º
30º1
3
160º
30º 21
23
hypotenuse
leg
leg
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
a
b
c
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.
First let’s look at the three basic functions.
SINECOSINE
TANGENT
They are abbreviated using their first 3 letters
c
a
hypotenuse
oppositesin
oppositec
b
hypotenuse
adjacentcos
adjacent
b
a
adjacent
oppositetan
The Trigonometric Functions
SINE
COSINE
TANGENT
SINE
Prounounced “sign”
Prounounced “co-sign”
COSINE
Prounounced “tan-gent”
TANGENT
Pronounced “theta”
Greek Letter
Represents an unknown angle
Pronounced “alpha”
Greek Letter α
Represents an unknown angle
Pronounced “Beta”
Greek Letter β
Represents an unknown angle
oppositehypotenuse
SinOpp
Hyp
adjacent
CosAdj
Hyp
TanOpp
Adj
hypotenuseopposite
adjacent
We could ask for the trig functions of the angle by using the definitions.
a
b
c
You MUST get them memorized. Here is a mnemonic to help you.
The sacred Jedi word:
SOHCAHTOA
c
b
hypotenuse
oppositesin
adjacentcos
hypotenuse
a
c opposite
tanadjacent
b
a
opposite
adjacent
SOHCAHTOA
It is important to note WHICH angle you are talking about when you find the value of the trig function.
a
bc
Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so
222 cba Let's choose: 222 5 43 3
45
sin = Use a mnemonic and figure out which sides of the triangle you need for sine.
h
o5
3
opposite
hypotenuse
tan =
a
o3
4
opposite
adjacent
Use a mnemonic and figure out which sides of the triangle you need for tangent.
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.
3
45
Oh, I'm
acute!
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.Sevenold horsesCrawled a hill To our attic..
Old Hippie
Old Hippie
SomeOldHippieCameAHoppin’ThroughOurApartment
SOHCAHTOA
Old Hippie
Old Hippie
SinOppHypCosAdjHypTanOppAdj
Other ways to remember SOH CAH TOA1.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 TOA1.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.
A
Trigonometry Ratios
Tangent A =
opposite
adjacent
Sine A =
opposite
hypotenuse
Cosine A =
adjacent
hypotenuse
Soh Cah Toa
14º
24º
60.5º
46º 82º
1.9 cm
7.7 cm
14º
1.9
7.7
0.25 Tangent 14º
0.25
The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side.
oppositeadjacent
hypotenuse
3.2 cm
7.2 cm24º
3.2
7.2
0.45 Tangent 24º
0.45
Tangent A =
opposite
adjacent
5.5 cm
5.3 cm
46º
5.5
5.3
1.04 Tangent 46º
1.04
Tangent A =
opposite
adjacent
6.7 cm
3.8 cm
60.5º
6.7
3.8
1.76
Tangent 60.5º
1.76
Tangent A =
opposite
adjacent
As an acute angle of a triangle approaches 90º, its tangent
becomes infinitely large
Tan 89.9º = 573
Tan 89.99º = 5,730
Tangent A =
opposite
adjacent
etc.
very large
very small
Since the sine and cosine functions alwayshave the hypotenuse as the denominator,
and since the hypotenuse is the longest side,these two functions will always be less than 1.
Sine A =
opposite
hypotenuse
Cosine A =
adjacent
hypotenuse
ASine 89º = .9998
Sine 89.9º = .999998
3.2 cm7.9 cm
24º
9.7
2.3
0.41 Sin 24º
0.41
Sin α = hypotenuse
opposite
5.5 cm
7.9 cm
46º
9.7
5.5
0.70 Cos 46º
0.70
Cosine β = hypotenuse
adjacent
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?
18º
x
After 60 sec., at 240 mph, the plane has traveled 4 miles
4
18º
x4
opposite
hypotenuse
SohCahToa
Sine A =
opposite
hypotenuse Sine 18 =
x
4
0.3090 =
x
4
x = 1.236 milesor
6,526 feet
1
Soh
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?
21º14.3
x
Tan 21 =
x
14.30.3839 =
x
14.3
x = 5.49 miles = 29,000 feet
1
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 theswimmer to the shore?
18º
150
Tan 18 =
x
150
x
0.3249 =
150
x
0.3249x = 150
0.3249 0.3249
X = 461.7 ft1
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?
x
60
120
Tan x =
60
120Tan x = 0.5
Tan-1(0.5) = 26.6º
Solving a Problem withthe Tangent Ratio
60º
53 ft
h = ?
We know the angle and the We know the angle and the side adjacent to 60º. We want to side adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use thetangent ratio:tangent ratio:
ft 92353
531
3
5360tan
h
h
h
adj
opp
1
2 3
Why?
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?
50
71.5°
?
tan 71.5°
tan 71.5° 50
y
y = 50 (tan 71.5°)
y = 50 (2.98868)
149.4y ft
Ex.
Opp
Hyp
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?
200
x
Ex. 5
60°
cos 60°
x (cos 60°) = 200
x
X = 400 yards
Trigonometric Functions on a Rectangular Coordinate System
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.
ry
x
The adjacent side is xThe opposite side is yThe hypotenuse is labeled rThis is called a REFERENCE TRIANGLE.
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Ratios may be found by:
45 º
1
1
2Using ratios of special trianglesUsing ratios of special triangles
145tan2
145cos
2
145sin
For angles other than 45º, 30º, 60º you will need to use a For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)calculator. (Set it in Degree Mode for now.)
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